1. science
2. mathematics
3. algebra

CHAPTER
1
Introduction to Probability
Timing
Materials
Blackline Masters
15–20 min
• standard deck of playing cards
BLM 1–1 Chapter 1 Self-Assessment
BLM 1–2 Chapter 1 Literacy Strategy
Core Expectations
A1.1 recognize and describe how probabilities are used to represent the likelihood of a result of an
experiment (e.g., spinning spinners; drawing blocks from a bag that contains different-coloured blocks;
playing a game with number cubes, playing Aboriginal stick-and-stone games) and the likelihood of
a real-world event (e.g., that it will rain tomorrow; that an accident will occur; that a product will be
defective)
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A1.2 describe a sample space as a set that contains all possible outcomes of an experiment, and
distinguish between a discrete sample space as one whose outcomes can be counted (e.g., all possible
outcomes of drawing a card or tossing a coin) and a continuous sample space as one whose outcomes can
be measured (e.g., all possible outcomes of the time it takes to complete a task or the maximum distance
a ball can be thrown)
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A1.3 determine the theoretical probability, Pi (i.e., a value from zero to one), of each outcome of a
discrete sample space (e.g., in situations in which all outcomes are equally likely), recognize that the sum
of the probabilities of the outcomes is 1 (i.e., for n outcomes, P1 + P2 + P3 + … + Pn = 1), recognize
that the probabilities Pi form the probability distribution associated with the sample space, and solve
related problems
A1.4 determine, through investigation using class-generated data and technology-based simulation
models (e.g., using a random-number generator on a spreadsheet or on a graphing calculator; using
dynamic statistical software to simulate repeated trials in an experiment), the tendency of experimental
probability to approach theoretical probability as the number of trials in an experiment increases (e.g.,
“If I simulate tossing two coins 1000 times using technology, the experimental probability that I calculate
1 than if I only simulate tossing
for getting two tails is likely to be closer to the theoretical probability of __
4
the coins 10 times”)
A1.5 recognize and describe an event as a set of outcomes and as a subset of a sample space, determine
the complement of an event, determine whether two or more events are mutually exclusive or nonmutually exclusive (e.g., the events of getting an even number or getting an odd number of heads from
tossing a coin repeatedly are mutually exclusive), and solve related probability problems [e.g., calculate
P(~A), P(A and B), P(A or B)] using a variety of strategies (e.g., Venn diagrams, lists, formulas)
A1.6 determine whether two events are independent or dependent and whether one event is conditional
on another event, and solve related probability problems [e.g., calculate P(A and B), P(A or B),
P(A given B)] using a variety of strategies (e.g., tree diagrams, lists, formulas)
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Chapter 1 Introduction to Probability • MHR
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Section
Learning Goals
1.1
• use probability to describe the likelihood of something occurring
• measure and calculate simple probabilities
1.2
• calculate theoretical probability
1.3
• recognize the difference between experimental probability and theoretical probability
1.4
• describe how an event can represent a set of probability outcomes
• recognize how different events are related
• calculate the probability of an event occurring
1.5
• describe and determine how the probability of one event occurring can affect the probability of another
event occurring
• solve probability problems involving multiple events
Notes
Use the visuals and introduction on pages 2–3 in Data
Management 12 to activate students’ prior knowledge
about the skills and processes that will be covered in
this chapter. Alternatively, you might want to use the
Prerequisite Skills on pages 4–5. Consider providing
students with a list of possible games for the Chapter
Problem. For board games, consider using Axis &
Allies, backgammon, bingo, Carcassonne, checkers,
chess, Civilization, Diplomacy, dominoes, Dungeons
& Dragons, Go, Illuminati, Kensington, mah-jong,
Monopoly, Risk, The Settlers of Catan, Scrabble®,
Tactics II, Ticket to Ride, Yahtzee, War; for card games,
consider using bridge, canasta, cribbage, hearts, euchre,
poker, and rummy. For electronic games, consider
Bejeweled® 3, Call of Duty®, Ninja Fishing, and
Candy Crush Saga.
Literacy Strategy
Students can use the Venn diagram as an effective
visual method of exploring non-mutually exclusive
events. Provide student pairs with a deck of playing
cards. Have them record the results using BLM 1–2
Chapter 1 Literacy Strategy. Alternatively, have
students use a table similar to the following:
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Total Red
Cards
Total Face
Cards
Total Red Cards
That Are Face Cards
Have student pairs create at least three more Venn
diagrams to sort cards that have a common outcome
into two groups. Alternatively, use an interactive
whiteboard and work as a class to place the cards in
a Venn diagram.
ELL
probability, bolt, predict, meteorology, atmosphere,
uncertainty, unknown outcomes, Venn diagram,
insurance underwriter, vehicle, die, spinner
Assessment Suggestions
Questioning
Assessment for Learning
• The Key Terms are introduced throughout the chapter.
Assess students’ understanding of the terms as they are
discussed.
• Have students develop a journal entry to explain what
they know about the Key Terms listed.
• Use the Prerequisite Skills to prepare students for
performing the calculations and organizing data in this
chapter.
• Have students complete the Before column of BLM 1–1
Chapter 1 Self-Assessment. Encourage them to refer to
this blackline master during the chapter.
• How likely is it that you will be struck by lightning?
• In what situations do you think the probability of being
struck by lightning would increase?
• How does a meteorologist use probability?
• Meteorologists predict the weather in the future. How
reliable do you think such forecasts are? Explain why.
• What do you know about probability?
• What subjects would you study in order to become an
insurance underwriter?
• What different random acts can you think of?
2 MHR • Chapter 1 Introduction to Probability 978-1-25-907746-3
Planning Chart
Section/Timing
Prior Learning
Materials
Chapter 1
Opener
15–20 min
Prerequisite
Skills
45–60 min
1.1 Simple
Probabilities
75 min
• standard deck of
playing cards
Teacher’s Resource
Blackline Masters
BLM 1–1 Chapter 1
Self-Assessment
BLM 1–2 Chapter 1
Literacy Strategy
Students should be
able to
• convert fractions
to decimals and
percent
• write fractions in
lowest terms
• add and subtract
fractions
• write ratios and
express ratios in
lowest terms
• determine
outcomes
• organize and
analyse data
Go to www.mathontario.ca
Resource Links for web
links for the following:
• converting fractions,
decimals, and percent
• adding and subtracting
fractions
• solving ratios
• creating a graph online
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Students should be
able to
• write and compare
fractions in lowest
terms
• add and subtract
fractions
• convert fractions
to decimals and
percent
• round calculated
percent
• coloured counters
(e.g., tiles, cubes)
• paper bag or
envelope
• chart paper
• 1.1 Investigate Extend
Your Understanding
Fathom™ Activity
Go to www.mathontario.ca
Resource Links for web
links for the following:
• simple probabilities
• converting fractions to
percent
• subjective probability
• an online calculator
• probability exercises
• an experiment to toss
up four coins
BLM 1–1 Chapter 1
Self-Assessment
BLM 1–3 Chapter 1
Warm-Up
BLM 1–4 Section 1.1
Investigate
BLM 1–5 Section 1.1
Practice
Students should be
able to
• organize and
record data for
a probability
experiment
• determine the
sample space and
the favourable
outcomes for
a probability
experiment
• 2 standard dice
• Internet access or
print media with
examples of odds
Go to www.mathontario.ca
Resource Links for web
links for the following:
• virtual dice
• an online probability
tree calculator
• calculating theoretical
probability
• tree diagrams
• probability of an event
not happening
• odds
BLM 1–1 Chapter 1
Self-Assessment
BLM 1–3 Chapter 1
Warm-Up
BLM 1–6 Section 1.2
Investigate
BLM 1–7 Section 1.2
Practice
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1.2 Theoretical
Probability
75 min
Media Links
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Chapter 1 Introduction to Probability • MHR
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Teacher’s Resource
Blackline Masters
Section/Timing
Prior Learning
Materials
Media Links
1.3 Compare
Experimental
and
Theoretical
Probabilities
75–150 min
Students should be
able to
• calculate
experimental
probability
• determine
theoretical
probability
• 3 coins
• graphing calculator
with Probability
Simulation
application or
computer with
spreadsheet
software
• 2 dice
• graphing calculator
with Probability
Simulation
application or
computer with
Fathom™
• 1.3 Investigate
Interactive Activity
Go to www.mathontario.ca
Resource Links for web
links for the following:
• downloading apps on a
graphing calculator
• simulations using coins
and dice
• analysing binomial
probabilities
• experimental and
theoretical probability
BLM 1–1 Chapter 1
Self-Assessment
BLM 1–3 Chapter 1
Warm-Up
BLM 1–8 Section 1.3
Practice
1.4 Mutually
Exclusive and
Non-Mutually
Exclusive
Events
75–150 min
Students should be
able to
• differentiate
between
experimental
probability
and theoretical
probability
• use simulation
to carry out
probability
experiments
• standard deck of
playing cards
• Venn diagram
• chart paper
• moveable letters
(optional)
• 3 different coloured
highlighters
• coloured counters
(3 colours)
• 1.4 Example 4 Solution
Animation
Go to www.mathontario.ca
Resource Links for web
links for the following:
• creating Venn diagrams
using Microsoft Word
• card games
• mutually exclusive and
non-mutually exclusive
events
• rule of sum
• principle of inclusion
and exclusion
Master 1 Venn Diagram
Master 2 Frayer Model
BLM 1–1 Chapter 1
Self-Assessment
BLM 1–3 Chapter 1
Warm-Up
BLM 1–9 Section 1.4
Investigate
BLM 1–10 Section 1.4
Practice
Go to www.mathontario.ca
Resource Links for web
links for the following:
• compound events
• the fundamental
counting principle and
independent events
• dependent events
• conditional probability
BLM 1–1 Chapter 1
Self-Assessment
BLM 1–3 Chapter 1
Warm-Up
BLM 1–11 Section 1.5
Practice
1.5
Independent
and
Dependent
Events
75–150 min
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Students should be
able to
• multiply fractions
• calculate
probability of
mutually exclusive
and non-mutually
exclusive events
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•
die
coloured counters
bag
chart paper
sticky notes
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Section/Timing
Prior Learning
Materials
Media Links
Review, Test
Yourself,
and Chapter
Problem
Wrap-Up
75–150 min
Go to www.mathontario.ca
Resource Links for web
links for the following:
• blackjack and Yahtzee
• experimental and
theoretical probability
• conditional probability
Teacher’s Resource
Blackline Masters
BLM 1–1 Chapter 1
Self-Assessment
BLM 1–5 Section 1.1
Practice
BLM 1–7 Section 1.2
Practice
BLM 1–8 Section 1.3
Practice
BLM 1–10 Section 1.4
Practice
BLM 1–11 Section 1.5
Practice
BLM 1–12 Chapter 1
Test
BLM 1–13 Chapter 1
Blackline Master
Answers
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Chapter 1 Introduction to Probability • MHR
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Prerequisite Skills
Timing
45–60 min
Media Links
• www.mathontario.ca Resource Links: Fractions, decimals, and percent, add and subtract fractions,
ratio and proportion, bar graphs
Notes
Meeting Student Needs
Have students complete #1 to #12 before starting the
content of this chapter.
• Have students use their math journal to keep track of
the skills and processes that need attention. As they
work on the chapter, they can check off each item as
they develop the skill at an appropriate level.
• Encourage ELL students to use an electronic
dictionary.
Method 1: Have students complete all Prerequisite
Skills questions before starting the chapter.
Method 2: Have students complete portions of the
Prerequisite Skills before they start on various sections
of the chapter. Refer to the Prior Learning column of
the Planning Chart for this chapter to identify topics
required for each section.
Method 3: Have students work in pairs. Assign each
Prerequisite Skills topic to a pair of students. Have
pairs of students prepare an exemplar of their assigned
skill and present it to the class.
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Skill/Concept
Meeting Student Needs
Fractions,
Decimals, and
Percent
• For reinforcement of converting fractions, decimals, and percent, go to www.mathontario.ca
Resource Links.
• For reinforcement of adding and subtracting fractions, go to www.mathontario.ca Resource
Links.
• A common error is to add the numerators and the denominators. Remind students that the
denominator stays the same.
Ratio and
Proportion
• For reinforcement of writing and solving ratios, go to www.mathontario.ca Resource Links.
Randomization
• Review some examples of random and non-random events.
Playing Cards
and Dice
• Allow students to practise using cards and dice.
Organizing,
Presenting, and
Analysing Data
• Allow students to practise creating graphs online. Go to www.mathontario.ca Resource Links.
6 MHR • Chapter 1 Introduction to Probability 978-1-25-907746-3
1.1
Timing
75 min
Learning Goal
Simple Probabilities
Materials
Media Links
• coloured counters • Fathom™ Activity: 1.1 Investigate Extend
(e.g., tiles, cubes)
Your Understanding
• paper bag or
• www.mathontario.ca Resource Links:
envelope
Probability, fractions to percent,
• chart paper
probability calculator, subjective
probability, probability exercises, tossing
coins experiment
Specific
Expectation
use probability
to describe
the likelihood
of something
occurring
A1.1
measure and
calculate simple
probabilities
A1.2
Blackline Masters
BLM 1–1 Chapter 1 Self-Assessment
BLM 1–3 Chapter 1 Warm-Up
BLM 1–4 Section 1.1 Investigate
BLM 1–5 Section 1.1 Practice
Sample Success Criteria
•
•
•
•
•
I can define probability.
I can identify an outcome.
I can describe the meaning of experimental probability.
I can explain what a probability means.
I can explain why experimental probability is not always accurate for making
predictions.
• I can use terms that mean the same as experimental probability.
• I can identify subjective probability.
•
•
•
•
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I can describe a sample space.
I can explain why the sum of probabilities is 1.
I can identify a discrete sample space.
I can determine probabilities based on experiments related to spinners, counters,
and games.
• I can apply experimental probability to calculate probabilities of real-world events.
• I can apply subjective probability to calculate probabilities of real-world events.
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Minds On...
Notes
ELL
Provide student pairs with a paper bag or envelope
that contains 10 coloured counters. Ask them to guess
how many of each coloured counter is in the bag.
Alternatively, have students use the photo to help
think of a method for determining how many of each
coloured counter is in the paper bag. The purpose is for
students to begin thinking about probability.
likelihood, estimate, mathematical processes, prediction,
uncertain
Assessment Suggestions
Questioning
Assessment for Learning
• BLM 1–3 Chapter 1 Warm-Up provides a
reactivation of prior learning skills to be used
at the beginning of each section. Have each
student complete the section 1.1 questions.
•
•
•
•
How many blue (red, yellow) counters do you think are in the bag?
What is the likelihood that one out of three counters will be blue?
What is the probability that the sun will rise tomorrow morning?
What is the probability that there will be a snow day in August?
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Chapter 1 Introduction to Probability • MHR
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Action!
Investigate Experimental Probability Notes
Meeting Student Needs
The Investigation has students carry out trials using
counters. Have students work in pairs. Direct students’
attention to the Literacy Link that explains at
random. Refer students to the Fathom™ activity called
Activity: 1.1 Investigate Extend Your Understanding,
which allows them to create a mystery bag and
randomly sample without replacement. As a class,
discuss students’ answers to Reflect and Extend Your
Understanding.
• Provide students with BLM 1–4 Section 1.1
Investigate, which provides a table for their answers.
• Have students use their math journal to define
probability, outcome, experimental probability, and
random in their own words. Ask them to provide an
example of each.
• For reinforcement of simple probabilities, refer
students to www.mathontario.ca Resource Links.
ELL
trials
Assessment Suggestions
Questioning
Assessment as Learning
• Listen as students discuss their answers to the Reflect
and Extend Your Understanding questions.
• When you flip a coin, what is the probability of the coin
landing heads?
• When you flip a coin, what is the probability of the coin
landing tails?
• When you roll a die, what is the probability of rolling a 1?
• When you roll a die, what is the probability of rolling a 6?
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Investigate Experimental Probability Answers
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3. Answers may vary. Example:
a)
Outcomes
Colour
Trial 1
Trial 2
Trial 3
Trial 4
Trial 5
Trial 6
Trial 7
Trial 8
Trial 9
Trial 10
Total
Blue (B)
|
|||| ||
|||| |||
||
|||| |
||||
|||
||||
|||
|||
42
Red (R)
|||| |
||
||
|||| ||
|||
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||
||||
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38
Yellow (Y)
|||
|
|
|
|||
||||
||
||||
20
b) The experimental probability of drawing blue is
42 = ___
21
P(B) = ____
100 50
The experimental probability of drawing red is
19
38 = ___
P(R) = ____
100 50
The experimental probability of drawing yellow is
20 = __
1
P(Y) = ____
100 5
I predict 5 blue, 3 red, and 2 yellow counters in
my partner’s bag. Or, I predict 4 blue, 4 red, and
2 yellow counters.
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c) I am not confident with my prediction. Since I
replace the counter in the bag after each draw, there
is a possibility I will draw the same counter from
the bag more than once in any of the 10 trials.
4. Answers may vary. Example:
a) In my partner’s mystery bag, there were 5 blue
counters, 3 red counters, and 2 yellow counters.
My prediction was very close.
b) No. Some predictions were very close. Others
were not very close. Some people who made the
same prediction as I did got 4 blue, 4 red, and 2
yellow counters.
5. Answers may vary. Example:
a) Making predictions based on the results of a low
number of trials will be less accurate than those
based on the results of a large number of trials.
b) To improve the accuracy of experimental
probability, carry out more trials (so that the
experimental probabilities will get closer to the
theoretical probabilities).
6. The experimental probability values would be very
accurate. Example: Since none of the 10 counters
drawn would be replaced before drawing the next
counter, the outcomes in each of the trials would be
the same and the experimental probabilities for each
colour in each trial would be the same.
Example 1 Notes
Meeting Student Needs
The example shows students how to calculate
experimental probability. Consider giving students a
choice of working on one of two parallel tasks that are
similar to Example 1. After they complete their work,
as a class compare the results of the parallel tasks and
the example. As a class, discuss how the spinner is
related to a pie graph and their advantage over a table.
Have students explain why the sum of probabilities will
always equal 1.
• Encourage students to visualize and verbalize what
the spinner might look like.
• Have students make a spinner and use it to carry out
trials and then calculate the experimental probability
of the spinner landing on each colour.
• For reinforcement of converting fractions to percent,
refer students to www.mathontario.ca Resource Links.
• Allow students who have difficulty calculating
experimental probability to use the online calculator at
www.mathontario.ca Resource Links.
• Have students use their math journal to record the
formula for calculating experimental probability.
Have them describe the formula in words. Have
students explain sum of probabilities.
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Example 2 Notes
Meeting Student Needs
The example shows a real-world application of
experimental probability. Have students create a pie
graph based on the statistical probabilities. Discuss
what the graph shows. As a class, have students share
other scenarios in which experimental probability may
be useful.
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• Consider having student pairs create their own realworld application and present the solution to the class.
• The Your Turn is similar to the example. Have
students refer to the example solution.
ELL
lunch rush, market researcher, cable, satellite, antenna
Example 3 Notes
Meeting Student Needs
Use the visual to explain that subjective probability
can be expressed on a scale from 0 to 1. Invite students
to share a scenario from their own experience. The
example shows how to estimate subjective probability.
Have students work with a partner to determine and
justify the subjective probability for each event. As a
class, discuss the justification for each probability.
• Have students develop and justify three different
scenarios involving subjective probability. Have them
present the results to the class.
• In their math journal, have students use their own
words to define and give an example of subjective
probability.
• For reinforcement of subjective probability, refer
students to www.mathontario.ca Resource Links.
ELL
subjective, scenario, shaker
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Chapter 1 Introduction to Probability • MHR
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Assessment Suggestions
Questioning
Assessment as Learning
• Encourage students to describe the concepts from the
examples in their math journal.
• Have students do the Example 1 Your Turn. Check that
students
− can determine total number of outcomes
− can calculate experimental probability as a fraction,
decimal, and percent
− can determine the sum of the probabilities
− can design a spinner that shows possible outcomes
− can determine if there is a fifth colour
• Have students do the Example 2 Your Turn. Check that
students
− can determine total number of outcomes
− can calculate experimental probability as a fraction,
decimal, and percent
− can apply reasoning to predict how results change
over time
• Have students do the Example 3 Your Turn. Check that
students
− can estimate subjective probability
− can justify their estimates
• Have students complete the related parts of BLM 1–1
Chapter 1 Self-Assessment.
• Students may benefit from peer assessment. Have them
work in a small group or in pairs.
Reasoning and Proving
Reflecting
• How do you calculate the experimental probability for each
television service?
• Who might be interested in the results? Why?
• What do you predict about the use of each television service
in five years?
• What is the difference between experimental probability
and subjective probability?
• What is the sum of probabilities for subjective probability?
How do you know?
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Mathematical Process
Problem Solving
• How do you determine the total number of outcomes for an
event?
• How do you determine the total number of favourable
outcomes for an event?
• How do you calculate the experimental probability of an
event?
• What is the sum of the probabilities of a probability
experiment in which there are n outcomes?
Question(s)
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Selecting Tools and Computational Strategies
Example 1 Your Turn part b) and c)
Example 3 Your Turn
Example 2 Your Turn part c)
Example 1 Your Turn part a)
Example 2 Your Turn part a)
Connecting
Representing
Example 1 Your Turn part b)
Communicating
Example 2 Your Turn part c)
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Consolidate and Debrief
Notes
Meeting Student Needs
As a class, review the Key Concepts. Then assign all the
Reflect questions before moving on to the exercises.
The exercise questions are contextual and include a
variety of styles and processes. Most longer questions
are scaffolded to help all levels of students. Consider
assigning both Practise questions and at least #4, #6, #8,
#10, and #12 of the Apply questions. Assign the Extend
questions to students who need a challenge. Treat #5
as a parallel task by having students create solutions
for similar questions that have different numbers of
coloured regions on the spinner or different restrictions
than the original question. Question #13 is an
Achievement Check. It includes an entry point in part
a) and communication in parts c) and d). Use #13 as an
assessment of learning. Consider allowing students to
use #17 as an alternative to the Chapter Problem.
• Have students work in pairs to answer the Reflect
questions.
• Direct students to the Literacy Link for R3. Ask what
other important words related to probability might be
used as hashtags.
• For #7, direct students to the Literacy Link
explaining PoP.
• For the Apply questions, consider having students
work in small groups to answer specific questions.
After students have completed their solutions, invite
volunteers to present their solutions to the class.
• For #11, extend the question by allowing students to
choose a favourite musician or actor and estimate the
subjective probability that he or she will win an award
(e.g., the Grammys, the Junos, the Oscars, the Actra
Awards).
• Question #12 and #14 are thinking questions and
require students to justify their solution. Look for the
depth of their justification.
• For #15, as an alternative to a graphing calculator,
students can roll an 8-sided die 20 times to generate
random numbers. Similarly, for #16 have students roll
a 4-sided die 10 times to generate random numbers.
• Assign #17 and #18 to students who need an extra
challenge.
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free throw, cone, pitcher, fast ball, curveball, knuckle ball,
Stanley Cup playoffs, Olympics, frequency
Assessment Suggestions
Questioning
Assessment of Learning
• Have students write a
response to R1 in their math
journal.
• Use students’ response to R2
as an exit ticket from class.
• Have students write a
response to R3 on chart
paper. Then have students
do a gallery walk of the
posted results.
• What is an example of experimental probability?
• What is the experimental probability of tossing 2 coins 20 times and getting heads once?
• Would the experimental probability of tossing 2 coins 100 times and getting heads once
be more or less accurate than tossing 2 coins 20 times and getting heads once? Explain
your thinking.
• What is an example of subjective probability?
• How are experimental probability and subjective probability the same? How are they
different?
• What is the probability that it will snow in Toronto in July? What is the probability that it
will not snow then in Toronto?
• What does an outcome of 0 represent? What does an outcome of 1 represent?
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Chapter 1 Introduction to Probability • MHR
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Learning Goal
Meeting Student Needs
use probability to describe the likelihood of something
occurring
• For probability exercises, go to www.mathontario.ca
Resource Links.
• For extra practice, have students complete BLM 1–5
Section 1.1 Practice.
measure and calculate simple probabilities
• Students may incorrectly record the results of an
experiment and get incorrect data. Remind them that
accuracy is important.
• For an experiment that allows users to toss up to four
coins and generate results, go to www.mathontario.ca
Resource Links.
• For extra practice, have students complete BLM 1–5
Section 1.1 Practice.
Mathematical Process
Question(s)
Problem Solving
#7, #13, #14
Reasoning and Proving
#3, #11, #12, #15, #18
Reflecting
R2, #6–#8
Selecting Tools and Computational Strategies
#15–#17
Connecting
#7
Representing
R3, #5, #9
Communicating
R1, #4, #10
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Achievement Check Sample Solution
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Frequency
13. a)
80
40
0
Red
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Green Orange
Blue
Colour
b) The total number of trials is
n(T) = 22 + 75 + 64 + 39, or 200.
n(R)
n(G)
P(R) = ____
P(G) = ____
n(T)
n(T)
22
____
____
=
= 75
200
200
= 0.11
= 0.375
n(B)
n(O)
P(B) = ____
P(O) = _____
n(T)
n(T)
64
____
____
=
= 39
200
200
= 0.32
= 0.195
12 MHR • Chapter 1 Introduction to Probability 978-1-25-907746-3
The experimental probability of drawing a red
counter is 11%.
The experimental probability of drawing a green
counter is 37.5%.
The experimental probability of drawing an
orange counter is 32%.
The experimental probability of drawing a blue
counter is 19.5%.
c) The number of counters must be a whole
number. Currently two of the probabilities are
not whole number percents. There cannot be 100
counters, because you cannot have 37.5 or 19.5
counters. However, multiplying by two eliminates
this problem. Then, there are 22 red, 75 green,
64 orange, and 39 blue counters, for a total of
200 counters.
d) Yes. The answer to part c) could be incorrect.
Since these are experimental probabilities, there
could be any number of counters of each colour.
1.2
Timing
75 min
Theoretical Probability
Materials
• 2 standard dice
• Internet access or
print media with
examples of odds
Media Links
• www.mathontario.ca Resource Links:
Virtual dice, probability tree calculator,
calculating theoretical probability,
tree diagrams, probability of an event
not happening, odds, probability tree
diagrams
Specific
Expectation
Learning Goal
Blackline Masters
BLM 1–1 Chapter 1 Self-Assessment
BLM 1–3 Chapter 1 Warm-Up
BLM 1–6 Section 1.2 Investigate
BLM 1–7 Section 1.2 Practice
Sample Success Criteria
calculate theoretical
probability using Venn
diagrams and tree
diagrams
A1.3
•
•
•
•
•
•
•
I can define theoretical probability.
I can give another name for theoretical probability.
I can distinguish between an event and a sample space.
I can use a Venn diagram to organize the outcomes of a sample space.
I can use a tree diagram to organize the outcomes of a sample space.
I can apply Venn diagrams to solve theoretical probability problems.
I can apply tree diagrams to solve theoretical probability problems.
calculate theoretical
probability using the
complement of an
event
A1.3
• I can define the complement of an event.
• I can apply the complement of an event to calculate theoretical probability.
apply probability to
calculate odds
A1.3
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• I can explain what is meant by odds.
• I can describe the difference between odds in favour and odds against.
• I can apply probability methods to calculate odds.
Minds On...
Notes
ELL
Use the introductory photo as a springboard for
students to brainstorm board games they are familiar
with that involve dice. Have students work in pairs to
discuss the questions in the Minds On before discussing
the answers as a class. You might challenge students
to organize the data for the sum of two dice before
proceeding to the Investigation. The purpose is for
students to begin thinking about theoretical probability.
strategy
978-1-25-907746-3
Chapter 1 Introduction to Probability • MHR
13
Assessment Suggestions
Questioning
Assessment for Learning
• Have students work in pairs to brainstorm ideas.
• Have students complete the section 1.2 questions on
BLM 1–3 Chapter 1 Warm-Up.
• When rolling a pair of dice, what sum can be rolled in the
greatest number of ways?
• When rolling a pair of dice, what sum can be rolled in the
least number of ways?
• When recording the sums of two dice, is the combination
1, 2 different from the combination 2, 1? Explain.
Action!
Investigate Outcomes and Events Notes
Meeting Student Needs
The Investigation has students analyse the possible
outcomes when two dice are thrown. Have students use
the table provided in step 1 to answer the questions.
Alternatively, provide students with two standard dice
to help them list all possible outcomes on BLM 1–6
Section 1.2 Investigate, and then answer step 1a) and
b) to step 5. As a class, discuss students’ answers to
Reflect and Extend Your Understanding.
• Have students work in pairs or small groups to
complete the activity.
• Provide a photocopy of the table in step 1 to students
who have trouble tracking favourable outcomes. Have
students use a highlighter to highlight the outcomes
on the copy.
• Have students use their math journal and define
theoretical probability, sample space, and event in their
own words. Encourage them to provide an example
for each term.
• For step 5, challenge students to compare theoretical
probability to experimental probability. Students
could roll two 8-sided dice 20 times and analyse the
sums. Then have them perform the same experiment
using Excel (if available) for a larger sample size
(possibly 200 times). For virtual dice, refer students to
www.mathontario.ca Resource Links.
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Assessment Suggestions
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Assessment as Learning
• Listen as students discuss their answers to the Reflect
and Extend Your Understanding questions.
Questioning
• What outcomes for sums have equal theoretical
probabilities?
• Do the probabilities for each sum change when an outcome
such as 1, 2 is treated the same as an outcome of 2, 1?
Explain.
14 MHR • Chapter 1 Introduction to Probability 978-1-25-907746-3
Investigate Outcomes and Events Answers
1. a) The sum that has the greatest theoretical
probability is 7. The value of this probability is
6 = __
1.
P(the sum is 7) = ___
36 6
b) The sums that have the lowest theoretical
probability are 2 and 12. The value
of the probability that the sum is 2 is
1 and the value of the
P(the sum is 2) = ___
36
probability that the sum is 12 is
1 .
P(the sum is 12) = ___
36
2. The probability of rolling a 9 or greater is
5 .
10 = ___
P(rolling a 9 or greater) = ___
36 18
3. Answers may vary. Example:
a) What is the probability of rolling a 5 or less?
b) The probability of rolling a 5 or less is
10 = ___
5 .
P(rolling a 5 or less) = ___
36 18
4. Answers may vary. Example: The table listed all
possible outcomes so it could be readily used to
count the number of theoretical probabilities for a
given probability problem.
Example 1 Notes
5. a) The theoretical probability of rolling a sum of
2 would decrease. The number of ways that the
event can occur is n(A) = 1 and the total number
of possible outcomes in the sample space is
n(S) = 64. The probability of rolling a sum of 2
n(A) 1
. The theoretical probability of
will be _____ = ___
n(S) 64
1 .
rolling a sum of 2 using 6-sided dice is ___
36
b) The theoretical probability of rolling a sum of
9 would increase. The number of ways that the
event can occur is n(A) = 8 and the total number
of possible outcomes in the sample space is
n(S) = 64. The probability of rolling a sum of 9 will
n(A)
8 = __
1 . The theoretical probability of
be ____ = ___
n(S) 64 8
4 = __
1.
rolling a sum of 9 using 6-sided dice is ___
36 9
c) The theoretical probability of rolling doubles
would decrease. The number of ways that the
event can occur is n(A) = 8 and the total number
of possible outcomes in the sample space is
n(S) = 64. The probability of rolling doubles will
n(A)
8 = __
1 . The theoretical probability of
be ____ = ___
n(S) 64 8
6 = __
1.
rolling doubles using 6-sided dice is ___
36 6
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EM
Meeting Student Needs
Introduce different ways of representing a sample space
(i.e., tree diagrams, Venn diagrams, set notation). Direct
students to the Literacy Link that explains set notation.
The example shows using a tree diagram to organize
possible outcomes. Have students create the tree
diagram for the example. Emphasize the importance of
labelling each branch. Model how to read the sample
space and record it.
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• Have students recreate the tree diagram using colour
instead of words to show all possible outcomes. Have
them verbalize the outcomes for each branch.
• For reinforcement, have students create their own
scenario, draw the tree diagram, and represent the
sample space. Have them exchange scenarios with a
classmate and create a tree diagram and sample space
for each other’s scenario.
• Provide a sample tree diagram for a similar scenario.
Have students identify the sample space and
determine the theoretical probability of drawing
specified combinations.
• In their math journal, have students use their own
words to define and give an example of set notation.
• Challenge students to use an online probability tree
calculator at www.mathontario.ca Resource Links.
• The Your Turn is similar to the example. Encourage
students to use the example as reference.
ELL
tree diagram, combination
978-1-25-907746-3
Chapter 1 Introduction to Probability • MHR
15
Example 2 Notes
Meeting Student Needs
Introduce the term complement and direct students
to the Literacy Link that explains the event A. The
example shows how to determine the probability of
a complement. It also shows how to represent the
relationship between A and A in a Venn diagram. Have
students verbalize the meaning of event A and A in
terms of the Battleship scenario.
• Have students use their math journal and define and
provide an example of complement.
• Some students may benefit from seeing an actual
Battleship board to determine the solution for part a).
• Have students use their math journal to describe how
A and A are complements using words and a Venn
diagram.
• Challenge students to represent the Your Turn in a
Venn diagram and describe it.
• Provide a different configuration of ships on a
Battleship board. Challenge students to answer parts
a) and b) using the configuration.
ELL
opponents, location, efficient
Example 3 Notes
Meeting Student Needs
The example illustrates how to calculate odds in favour
and odds against based on subjective probabilities.
Use the information about odds to describe odds in
favour and odds against. As a class, have students
share examples that they are familiar with before they
work through the example. For the Your Turn, remind
students to reduce ratios to lowest terms.
• For part b), direct students to the Literacy Link for
an explanation of OFSAA.
• Have students use their math journal and define and
provide an example of odds in favour and odds against
for a sport of their choice. Have students write the
ratio for each.
• Have students use their math journal to describe two
sets, A and A, that are complements of each other
using words and a diagram. Have them calculate
the odds in favour and the odds against an event
occurring.
• Coach students who have difficulty with converting
percent to decimals and fractions.
• The Your Turn is similar to the example. Tell students
to refer to the solution to the example.
• Challenge students to use the scenario in Example 2
and calculate the odds in favour of randomly hitting
a ship on the first guess and the odds against hitting a
ship on the first guess.
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sports analyst, tournament, hockey analyst, sports
journalist
16 MHR • Chapter 1 Introduction to Probability 978-1-25-907746-3
Assessment Suggestions
Questioning
Assessment as Learning
• Encourage students to describe the concepts from the
examples in their math journal.
• Have students do the Example 1 Your Turn. Check that
students
− can create a tree diagram with appropriate branches
and labels
− can count the number of possible outcomes
− can use the tree diagram to determine theoretical
probability
• Have students do the Example 2 Your Turn. Check that
students
− can identify complementary events
− can determine the theoretical probability of randomly
choosing an event
− can determine the theoretical probability of randomly
not choosing an event
− can apply complementary reasoning
• Have students do the Example 3 Your Turn. Check that
students
− can convert a percent to a decimal
− can determine the probability of an event not
occurring
− can determine the probability of an event occurring
and the probability of an event not occurring
− can use the definition of odds to determine the odds
in favour of an event and the odds against an event
• Have students complete the related parts of BLM 1–1
Chapter 1 Self-Assessment.
• Students may benefit from peer assessment. Have them
work in a small group or in pairs.
•
•
•
•
•
How many branches will there be for hat?
How many branches will there be for gloves?
How many possible outcomes are there?
What is the sample space?
What formula represents the probability that the hat and
gloves are the same colour?
• What are the complementary events?
• How could you represent the relationship between the
complementary events visually?
• What is the theoretical probability of randomly choosing a
blueberry muffin?
• How can you apply complementary reasoning to determine
the theoretical probability of not choosing a blueberry
muffin? What relationship do you use?
• What relationship do you use to determine the probability
of not making the playoffs?
• What ratio can you use to determine odds in favour?
• What ratio can you use to determine odds against?
• If there is an 85% chance that the Toronto Maple Leafs will
win the Stanley Cup, what are the odds in favour of the
Toronto Maple Leafs winning the next Stanley Cup?
• If there is a 65% chance that the Ottawa Senators will win
the Stanley Cup, what are the odds against the Ottawa
Senators winning the Stanley Cup?
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Mathematical Process
Question(s)
Problem Solving
Example 1 Your Turn
Example 2 Your Turn
Reasoning and Proving
Reflecting
Selecting Tools and Computational Strategies
Example 1 Your Turn
Example 3 Your Turn part a)
Connecting
Example 3 Your Turn part b)
Representing
Example 1 Your Turn
Communicating
Example 2 Your Turn
978-1-25-907746-3
Chapter 1 Introduction to Probability • MHR
17
Consolidate and Debrief
Notes
Meeting Student Needs
As a class, review the Key Concepts. Then assign the
Reflect questions before moving on to the exercises.
Identify any difficulties students have and provide
remediation before proceeding. Consider allowing
students to work with a partner but have them record
their own solutions. Assign all the Practise questions
and at least #4 to #6 and #8 to #10 in the Apply
questions. Assign the Extend questions to students
who need a challenge. Consider using #10, which is
an Achievement Check, for assessment of learning
purposes. For #11, provide access to the Internet or
print media with examples of odds. For #11, as a class
discuss instances when the term odds is used incorrectly
in the media. For example, the media may express odds
as a probability (not as a ratio) and vice versa. You
might challenge students to research examples of odds
being used incorrectly and post the results.
• Use #1 to #3 to quickly assess students’ understanding
of concepts.
• For #1, remind students that there are 52 cards in a
deck, with 12 face cards.
• Questions #12, #13, and #17 are thinking questions
and require students to justify their solution. Look for
the depth of their justification.
ELL
toothpicks, learned behaviour
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Assessment Suggestions
Questioning
Assessment of Learning
• Have students write a response to R1 in their math
journal. Have them post their visual organizer in the
classroom. Then have students do a gallery walk of the
posted results.
• For R2, have students exchange their example of an
event and its complements with a classmate and
solve each other’s theoretical probabilities. Have them
compare the results.
• Have students use a Venn diagram to represent their
response to R3.
• If two standard dice are thrown, what is the theoretical
probability that the sum of the two dice will be 8?
• What is the sample space, in set notation, for 6 blue T-shirts,
4 red T-shirts, and 3 green T-shirts?
• If the probability of an event is 70%, what is the probability
of the complement of the event?
• What are odds often based on?
• What is an example of odds in favour of an event?
• What is an example of odds against an event?
• How do you express odds in favour or odds against?
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Learning Goal
Meeting Student Needs
calculate theoretical
probability using Venn
diagrams and tree
diagrams
• To calculate the theoretical probability of getting two heads when flipping coins,
students may flip two coins once, instead of one coin twice, and draw an incorrect tree
diagram. To help students, go to www.mathontario.ca Resource Links.
• For reinforcement of tree diagrams, go to www.mathontario.ca Resource Links.
• For extra practice, have students complete BLM 1–7 Section 1.2 Practice.
calculate theoretical
probability using the
complement of an event
• For reinforcement of finding the probability of an event not happening, go to
www.mathontario.ca Resource Links.
• For extra practice, have students complete BLM 1–7 Section 1.2 Practice.
apply probability to
calculate odds
• For reinforcement of odds in favour and odds against, go to www.mathontario.ca
Resource Links.
• For extra practice, have students complete BLM 1–7 Section 1.2 Practice.
18 MHR • Chapter 1 Introduction to Probability 978-1-25-907746-3
Mathematical Process
Question(s)
Problem Solving
#6, #10, #12
Reasoning and Proving
#8, #9, #13, #14, #17
Reflecting
#14
Selecting Tools and Computational Strategies
#4
Connecting
R2, #15
Representing
R1, #5, #10, #16
Communicating
R1, R2, R3, #7, #11, #14, #15, #17
Achievement Check Sample Solution
10. a) Let C represent correct and W represent
incorrect.
Q1
Q2
Q3
C
Q4
C
W
C
C
C
W
W
C
C
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C
W
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C
C
W
W
C
W
C
W
W
C
C
b) all four correct: From the tree diagram, n(A) = 1
and n(S) = 16. So, the experimental probability
1 , or 0.0625, or
that Kwon gets all four correct is ___
16
6.25%.
exactly three correct: From the tree diagram,
n(A) = 4 and n(S) = 16. So, the experimental
probability that Kwon gets exactly three correct
4 , or 0.25, or 25%.
is ___
16
fewer than two correct: Assume that fewer than
two correct means 1 or 0 correct. From the
tree diagram, n(A) = 5 and n(S) = 16. So, the
experimental probability that Kwon gets fewer
5 , or 0.3125, or 31.25%.
than two correct is ___
16
not all incorrect: From the tree diagram,
n(A) = 15 and n(S) = 16. So, the experimental
15 ,
probability that Kwon gets not all incorrect is ___
16
or 0.9375, or 93.75%.
W
W
C
W
W
978-1-25-907746-3
Chapter 1 Introduction to Probability • MHR
19
1.3
Timing
75 min–150 min
Compare Experimental and
Theoretical Probabilities
Materials
• 3 coins
• graphing calculator with
Probability Simulation application
or computer with spreadsheet
software
• 2 dice
• graphing calculator with
Probability Simulation application
or computer with Fathom™
Learning Goal
recognize the
difference between
experimental
probability and
theoretical probability
Minds On...
Specific
Expectation
A1.4
Media Links
• www.mathontario.ca Resource
Links: Downloading apps on a
graphing calculator, simulations
using coins and dice, analysing
binomial probabilities,
experimental and theoretical
probability
• Interactive Activity: 1.3 Investigate
Blackline Masters
BLM 1–1 Chapter 1
Self-Assessment
BLM 1–3 Chapter 1
Warm-Up
BLM 1–8 Section 1.3
Practice
Sample Success Criteria
•
•
•
•
I can use a graphing calculator to simulate experimental probability.
I can use a spreadsheet to simulate experimental probability.
I can use Fathom™ to simulate experimental probability.
I can design and carry out a probability experiment using a variety of
methods.
• I can recognize the relationship between experimental and theoretical
probability for a large number of trials.
• I can explain the difference between experimental and theoretical
probability.
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Notes
Use the photo to generate a discussion about simulators
that students have heard about or experienced. For
example, they may have tried a motion simulator
at a museum or online. Ask them to share their
experience. Students may mention simulators used in
sports (e.g., golf), medicine (e.g., birthing simulator,
cardiopulmonary simulator), or operating machines or
equipment (e.g., learning to drive a car). The purpose
is for students to begin thinking about simulating
probability experiments.
20 MHR • Chapter 1 Introduction to Probability 978-1-25-907746-3
ELL
astronauts, flight simulator
Assessment Suggestions
Questioning
Assessment for Learning
• Have students complete the
section 1.3 questions on BLM 1–3
Chapter 1 Warm-Up.
•
•
•
•
•
•
What professions involve training using a simulator?
Why are simulators important for training people?
What kinds of simulator programs are available for use at home?
What kinds of simulator software are available online?
What type of technology can be used to simulate probability experiments?
What probability simulations on a programmed graphing calculator are you
familiar with?
Action!
Investigate 1 Three-Coin-Flip Simulation and
Investigate 2 Dice Simulation Notes
Meeting Student Needs
Students should complete the two Investigations. They
choose one of three methods for each Investigation.
Make sure that all three methods are represented so
that students can discuss the results for all methods as a
class. Both investigations could be set up using stations.
The Investigations could be done using technology such
as Gizmos® or graphing calculators. For instructions
about downloading apps on a graphing calculator, go to
www.mathontario.ca Resource Links.
In the first Investigation, students compare experimental
probability to theoretical probability when three coins are
flipped. The activity can be completed with or without
technology. Discuss the Processes (Selecting Tools and
Computational Strategies and Representing) question
(the sum of any row of columns A to C represents
the number of heads that occur when three coins are
flipped). Ask students to respond to the Processes
(Reflecting) question. (It is necessary to go to row 101 so
that there are 100 trials. Row 1 lists the four favourable
outcomes for the sample space.)
• For Investigate 1, consider allowing students to create
a tree diagram for step 1 on a SMART Board.
• Allow students to work in pairs or small groups.
Students should each record their own results. Have
each group present their method to the class.
• Have students work in groups of three with each
member choosing a different method. Provide time
for students to compare their results.
• For an analysis of outcomes when two, three,
and four coins are tossed, refer students to
www.mathontario.ca Resource Links
• Invite students to try the binomial coin experiment at
www.mathontario.ca Resource Links (see the dropdown menu for binomial coin toss).
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tedious, fair coins, simulation, tallied, bar graph,
spreadsheet, clustered column, attributes, workspace
In the second Investigation, students compare
experimental probability to theoretical probability
when two dice are tossed. For a dice simulation,
refer to the interactive activity called 1.3 Investigate.
Alternatively, students can roll dice on a SMART Board.
Use the Literacy Link about histograms to point out the
difference between a bar graph and a histogram. After
each Investigation, as a class discuss the results and the
methods students preferred and why. It is important
for students to complete the Reflect and Extend Your
Understanding questions and discuss the answers as
a class. Consider using the answers for assessment as
learning.
978-1-25-907746-3
Chapter 1 Introduction to Probability • MHR
21
Assessment Suggestions
Questioning
Assessment as Learning
• Observe the ability of each student to use the method
chosen to determine the theoretical probabilities for the
coin flip and the dice simulation.
• Listen as students discuss their answers to the Reflect
and Extend Your Understanding questions.
• How many branches will the tree diagram have?
• How are the theoretical probabilities for flipping one, two,
three, four, and five coins related?
• When is it useful to use technology-based simulations to
calculate theoretical probability?
Investigate 1 Three-Coin-Flip Simulation Answers
1.
h) The graph is very close to the theoretical
prediction.
First Coin Second Coin Third Coin Outcomes
H
HHH
H
H
HHT
T
HTH
H
HTT
T
T
H
H
THH
T
THT
H
TTH
T
T
TTT
T
There are 8 possible outcomes.
1
2. a) P(exactly three heads) = __
8
3
b) P(exactly two heads) = __
8
3
c) P(exactly one head) = __
8
1
d) P(no heads) = __
8
3. Method 1: Use a Graphing Calculator. Answers may
vary. Example:
d) The graph is similar to the theoretical prediction.
e) The graph is very close to the theoretical prediction.
Method 2: Use a Spreadsheet. Answers may vary.
Example:
f)
Frequency
400
Frequency
4
2
0
0
1
2
3
Number of Heads
g) The graph is similar to the theoretical prediction.
Frequency
40
30
20
10
0
0
1
2
Number of Heads
3
200
100
0
0
1
2
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Method 3: Combine Real Trials. Answers may vary.
Example:
0 ; P(1 head) = ___
2;
a) P(0 heads) = ___
10
10
4 ; P(3 heads) = ___
4;
P(2 heads) = ___
10
10
No, not very useful. Not all of the outcomes
occurred. The probabilities are not similar to
the theoretical probabilities.
b) The graph is similar to the theoretical prediction.
c) The graph is very close to the theoretical
prediction.
4. The statistical probabilities of the experiment become
very close to the theoretical probabilities of the
experiment as the total number of trials increases.
5. Answers may vary. Example: a) It would take more
trials for the statistical probabilities of the outcomes
to match the theoretical probabilities since there are
more outcomes.
b) The sample space has 25 = 2 × 2 × 2 × 2 × 2 = 32
outcomes. Step 1. Calculate theoretical probability
when 5 fair coins are flipped:
5;
1 ; P(exactly 4 heads) = ___
P(exactly 5 heads) = ___
32
32
10 ; P(exactly 2 heads) = ___
10 ;
P(exactly 3 heads) = ___
32
32
5 ; P(0 heads) = ___
1.
P(exactly 1 head) = ___
32
32
Step 2. Use a spreadsheet to simulate flipping
5 coins 1000 times.
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22 MHR • Chapter 1 Introduction to Probability 978-1-25-907746-3
Step 3. Record the experimental probability and
create a graph.
Method 2: Use Fathom™. Answers may vary.
Example:
d)
Frequency
of Sum
Frequency
300
200
100
2
1
0
0
1
2
4
3
Step 4. Summary: The graph of the experimental
probability is very close to the theoretical
probability.
Frequency of Sum
Frequency of Sum
1. There are 36 possible outcomes.
1 ; P(sum of 3) = ___
2;
P(sum of 2) = ___
36
36
3 ; P(sum of 5) = ___
4;
P(sum of 4) = ___
36
36
5 ; P(sum of 7) = ___
6;
P(sum of 6) = ___
36
36
5 P(sum of 9) = ___
4;
P(sum of 8) = ___
36
36
3 ; P(sum of 11) = ___
2;
P(sum of 10) = ___
36
36
1
P(sum of 12) = ___
36
2.
0
2
5
4
9
7
6
8
8
4
2
4
6
8
Sum
10
12
14
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500
400
300
200
100
11
10
14
12
0
3
12
f) I had to use 3200 cases for the bar graph to look
very similar to the theoretical probabilities.
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0
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6
8
Sum
20
Investigate 2 Dice Simulation Answers
4
4
e) The graph is similar to the theoretical prediction.
Number of Heads
6
2
5
Frequency of Sum
0
12
Sum
3. Method 1: Use a Graphing Calculator. Answers may
vary. Example:
c) As the number of trials increases by 50 at a
time, the bar graph representing the statistical
probabilities looks very similar to the theoretical
probabilities. After 401 trials, the total number of
rolls for a sum of 7 in the centre of the bar graph is
the most and the total number of rolls decreases to
the left and to the right to the least number for a
sum of 2 or a sum of 12, respectively.
2
4
6
8
Sum
10
12
14
Method 3: Combine Real Trials. Answers may vary.
Example:
0 ; P(sum of 3) = ___
2;
a) P(sum of 2) = ___
10
10
0 ; P(sum of 5) = ___
3;
P(sum of 4) = ___
10
10
1 ; P(sum of 7) = ___
1;
P(sum of 6) = ___
10
10
1 ; P(sum of 9) = ___
1;
P(sum of 8) = ___
10
10
0 ; P(sum of 11) = ___
1;
P(sum of 10) = ___
10
10
1 ; No, not very useful. Not all
P(sum of 12) = ___
10
of the sums were rolled. The probabilities are not
similar to the theoretical probabilities.
978-1-25-907746-3
Chapter 1 Introduction to Probability • MHR
23
b) The graph is similar to the theoretical prediction.
16
12
8
4
0
2
4
6
8
Sum
10
12
14
Frequency of Sum
c) The graph is similar to the theoretical prediction.
240
160
80
0
2
4
6
8
Sum
10
12
14
4. Answers may vary. Example: The statistical
probabilities of the experiment become very close to
the theoretical probabilities of the experiment as the
total number of trials increases.
5. Answers may vary. Example: I had to use 3200
cases for the bar graph to look very similar to the
theoretical probabilities. The “law of large numbers”
says that as the number of trials of a random process
increases, the percent difference between the
expected and actual values goes to zero.
Frequency of Sum
Frequency of Sum
20
6. Answers may vary. Example: The sample space would
increase from 36 to 36 × 6 = 216 outcomes. There are 16
sums that could be rolled from 3 to 18. Step 1. Calculate
the theoretical probability when 3 fair dice are rolled.
Divide the total number of ways to obtain each sum by
the total number of outcomes in the sample space.
3 ;
1 ; P(4) = P(17) = ____
P(3) = P(18) = ____
216
216
6 ; P(6) = P(15) = ____
10 ;
P(5) = P(16) = ____
216
216
15 ; P(8) = P(13) = ____
21 ;
P(7) = P(14) = ____
216
216
25 ; P(10) = P(11) = ____
27 .
P(9) = P(12) = ____
216
216
Step 2. Use Fathom™ to simulate rolling 3 fair dice
100 times. Then use Fathom™ to simulate rolling 3
fair dice 1000 times.
Step 3. Record the experimental probability and
create a graph.
80
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2
4
6
8
10 12
Sum
14
16
18
20
Step 4. Summary: The graph of the experimental
probability is very close to the theoretical probability.
The probabilities for the sums of 3 and for 18 are the
least probable. The sums exactly in the middle are
the most probable. This is similar to the experimental
probability when two dice are rolled.
Consolidate and Debrief
Notes
Meeting Student Needs
As a class, review the Key Concepts. Then have
students answer the Reflect questions, perhaps in
pairs. Assign both Practise questions and one of #3 or
#4, #5, #6, and one of #7 or #8 of the Apply questions.
Assign #9 and/or #10 to students who need a challenge.
Students will need technology tools for #3 to #7 and
#9 and #10. Consider which questions to assign for
homework to make the best use of the technology tools
available to students. Questions involving technology
tools not available at home could be assigned for
completion in class, perhaps in a collaborative setting.
Consider having students work in small groups to
answer #6, which is an Achievement Check, and use
the results for assessment of learning purposes. Assign
the Extend questions to students who need a challenge.
• Have students use #1 and #2 for self-assessment.
• For #3 and #4, challenge students to use a spreadsheet
or Fathom™ as an alternative to graphing calculators.
• For #5, students may choose to create a bar graph
using a spreadsheet program.
• For #10, have students use search terms for
online probability simulators such as those at
www.mathontario.ca Resource Links.
24 MHR • Chapter 1 Introduction to Probability 978-1-25-907746-3
ELL
physical materials, perfect predictor, trend behaviour
Assessment Suggestions
Questioning
Assessment of Learning
• Have students post the results for #6 in the classroom.
Provide an opportunity for students to do a gallery walk to
compare results.
• For R1, use a think-pair-share strategy.
• For R2, have students write a response in their math journal.
• How many possible outcomes are there for rolling four
dice?
• Why are the results of experimental probability different
from the results of theoretical probability?
• When do the results of experimental probability
approach the results of theoretical probability?
Learning Goal
Meeting Student Needs
recognize the difference
between experimental
probability and
theoretical probability
• A common error occurs when students fail to recognize the difference between
results from experimental probability and the actual theoretical probability of an
event. For reinforcement of experimental probability and theoretical probability, go to
www.mathontario.ca Resource Links.
• For extra practice, have students complete BLM 1–8 Section 1.3 Practice.
Mathematical Process
Question(s)
Problem Solving
#3
Reasoning and Proving
R2, #5
Reflecting
#8
Selecting Tools and Computational Strategies
#6, #9
Connecting
#4, #8
Representing
#7
Communicating
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Achievement Check Sample Solution
Theoretical Probability
6. a) The sample space for tossing two fair coins is
S = {HH, HT, TH, TT}.
1 ; exactly one head: __
1 ; two heads: __
1
b) no heads: __
4
4
2
c)
0.6
R1, #10
d)
e) Answers may vary. Example: The graphs look
the same. I ran 1500 trials to obtain the graphing
calculator screen in part d).
0.4
0.2
0
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0 heads 1 head 2 heads
Number of Heads
978-1-25-907746-3
Chapter 1 Introduction to Probability • MHR
25
1.4
Timing
75 min–150 min
Mutually Exclusive and
Non-Mutually Exclusive Events
Materials
• standard deck of
playing cards
• Venn diagram
• chart paper
• moveable letters
(optional)
• 3 different coloured
highlighters
• coloured counters
(3 colours)
Learning Goal
Specific
Expectation
Media Links
• www.mathontario.ca Resource Links:
Venn diagrams using Microsoft Word,
card games, mutually exclusive and
non-mutually exclusive events, rule
of sum, principle of inclusion and
exclusion
• Animation: 1.4 Example 4 Solution
Blackline Masters
Master 1 Venn Diagram
Master 2 Frayer Model
BLM 1–1 Chapter 1 SelfAssessment
BLM 1–3 Chapter 1 Warm-Up
BLM 1–9 Section 1.4
Investigate
BLM 1–10 Section 1.4 Practice
Sample Success Criteria
describe how an event
can represent a set of
probability outcomes
A1.5
• I can recognize an event as a set of outcomes.
• I can describe an event as a subset of a sample space.
recognize how
different events are
related
A1.5
• I can identify mutually exclusive events.
• I can identify non-mutually exclusive events.
• I can explain the difference between mutually exclusive and non-mutually
exclusive events.
• I can distinguish between mutually exclusive and non-mutually exclusive
events.
• I can explain the additive principle.
• I can identify when to use the additive principle.
• I can explain the principle of inclusion and exclusion.
• I can identify when to use the principle of inclusion and exclusion.
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calculate the
probability of an event
occurring
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A1.5
•
•
•
•
I can determine the probability of mutually exclusive events.
I can determine the probability of non-mutually exclusive events.
I can apply the additive principle to determine the probability of an event.
I can apply the principle of inclusion and exclusion to determine the
probability of an event.
• I can apply a variety of methods to determine the probability of an event
occurring.
26 MHR • Chapter 1 Introduction to Probability 978-1-25-907746-3
Minds On...
Notes
Use the photo as a starting point for a class discussion
about playing cards. Have students work in pairs to list
the card games that they are familiar with. Provide time
for students to discuss what makes each card game on
the list interesting to play. The purpose is for students to
think about how card games are related to probability.
Assessment Suggestions
Questioning
Assessment for Learning
• Have students contribute to a class list of card games
and post the list in the classroom. Have students do a
gallery walk.
• Have students complete the section 1.4 questions on
BLM 1–3 Chapter 1 Warm-Up.
•
•
•
•
•
What is your favourite card game?
What card games are played by only one person at a time?
What card games are played with more than one person?
Card games are games of chance. Do you agree? Why?
What is the probability of randomly drawing a queen from
a deck of cards?
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Action!
Investigate Counting Cards Notes
Meeting Student Needs
The Investigation has students compare the attributes
of cards in a hands-on activity. Make decks of playing
cards available. Provide BLM 1–9 Section 1.4
Investigate, which students can use to complete the
Venn diagrams. The activity leads to the introduction of
mutually exclusive events.
• Refer students to www.mathontario.ca Resource
Links to create Venn diagrams using Microsoft Word.
• Invite students to research card games played in
different countries. Refer to www.mathontario.ca
Resource Links.
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simultaneously
Assessment Suggestions
Questioning
Assessment as Learning
• Listen as students discuss their answers to the Reflect
and Extend Your Understanding questions.
• In a standard deck of playing cards, how many cards are
clubs? How many cards are hearts?
• In a standard deck of playing cards, how many cards are
clubs, face cards, or both?
978-1-25-907746-3
Chapter 1 Introduction to Probability • MHR
27
Investigate Counting Cards Answers
1. a) 13
b) 13
2. 26
3.
Clubs
Spades
A
2
A
3
4
2
3
4
5
6
7
5
6
7
8
9
10
8
9
10
J
Q
K
J
Q
K
4. 26
5. Yes, the answers are the same. Answers may vary.
Example: 13 clubs and 13 spades add to 26 cards
in total.
6. a) 13
b) 12
7. 25
8.
Diamonds
Face Cards
A
3
4
J
J
Q
K
5
6
7
Q
J
Q
K
K
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9
10
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2
8
9. 22
10. No, the answers are not the same. Answers may
vary. Example: For step 7, there are 13 diamonds
and 12 face cards for a total of 25 cards. For step
9, there are 3 cards that are both a diamond and a
face card. That accounts for the difference of 3. To
determine the total number of cards in the Venn
diagram, the 3 cards that are both a diamond and a
face card must be subtracted from the total number
of cards that are diamonds and face cards.
26 = __
1
11. a) P(club or spade) = ___
52 2 22 11
b) P(diamond or face card) = ___ = ___
52 26
c) Answers may vary. Example: For part a), I
counted 52 cards in the deck. I added the total
number of clubs (13) and the total number of
spades (13) to get a total of 26 cards that were
clubs or spades. I calculated the probability of
drawing either a club or spade by dividing the
number of cards that were clubs or spades by
the total number of cards. For part b), I counted
52 cards in the deck. I added the total number of
diamonds (13) and the total number of face cards
(12) to get a total of 25 cards that were diamonds
or face cards. I subtracted the number of cards
that were both a diamond and a face card (3) to
get a total of 22 cards that were a diamond or a
face card. I calculated the probability of drawing
a diamond or a face card by dividing the number
of cards that were a diamond or a face card by the
total number of cards.
Example 1 Notes
Meeting Student Needs
Before working through the example, discuss the
information about mutually exclusive events and ask
students for some examples. Method 1 shows how to
determine the probability of mutually exclusive events by
dividing the total number of favourable outcomes by the
total number of all possible outcomes. Method 2 shows
how to determine the probability of mutually exclusive
events by adding the probabilities of the favourable
outcomes. Provide students with Master 1 Venn
Diagram. Encourage students to summarize the two
methods in their math journal. Ask students to use both
methods for the Your Turn. After taking up the solutions
as a class, discuss the additive principle for mutually
exclusive events. Walk through the four steps that show
the relationship between Method 1 and Method 2 in
the example. Afterward, write each step on a separate
piece of chart paper and have students work in pairs to
organize the steps in order to prove the additive principle
(rule of sum) for mutually exclusive events.
• For tactile and visual learners, invite students to use
Master 1 Venn Diagram and moveable letters to
represent the sandwiches. Consider creating a parallel
task for students where they choose to calculate the
probability of randomly picking either a ham or an
egg salad sandwich, or the probability of picking
either a turkey or a chicken sandwich. Consider doing
the same for the Your Turn.
• In their math journal, have students record their own
definition and example for mutually exclusive events
and the rule of sum.
28 MHR • Chapter 1 Introduction to Probability 978-1-25-907746-3
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picnic basket, cooler
Example 2 Notes
Meeting Student Needs
This example shows how to apply the rule of sum for
mutually exclusive events. To reinforce the relationship
between the methods used in Example 1, consider
providing a parallel task by asking students to
determine the probability that Rolly will randomly pick
either a Connery or Dalton movie using either method
from Example 1. As a class, take up the solutions to
show that the results are the same.
• Refer students who may benefit from additional
examples of applying the rule of sum to
www.mathontario.ca Resource Links.
Example 3 Notes
Meeting Student Needs
Discuss the information about non-mutually exclusive
events and the principle of inclusion and exclusion
to introduce the example. Direct students to the
Literacy Link that explains union set and intersection
set. Note that the notation is not used in this chapter.
The example illustrates how to apply the principle of
inclusion and exclusion by determining the number
of favourable outcomes, and then determine the
probability of the favourable outcome by dividing by
the total number. You might divide students into three
groups: students who play basketball, those who play
volleyball, and those who play both sports. Have each
group create a Venn diagram on chart paper and then
use two steps to calculate the probability that a student
chosen at random plays basketball or volleyball. Outline
the steps as needed (apply the principle of inclusion and
exclusion by counting the number of students who play
basketball or volleyball, and divide by the total number
of students). As a class, discuss the posted solutions.
• For step 8, provide student pairs with a deck of
cards and ask them to create new scenarios for two
events where there is a common event. Have students
represent the principle of inclusion and exclusion
using Master 1 Venn Diagram.
• In their math journal, have students record a
definition and example for non-mutually exclusive
events.
• Have students use Master 1 Venn Diagram to
represent the principle of inclusion and exclusion
with an example.
• Challenge students to research union set and
intersection set and create a poster that includes
the notation and a description of each term. Have
students present their findings orally.
Example 4 Notes
Meeting Student Needs
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Discuss the introductory information about the
probability of two non-mutually exclusive events.
The example shows how to apply the formula for the
probability of two non-mutually exclusive events by
determining the probability of each event and then
subtracting the probability of both events happening
at the same time. Reinforce the usefulness of a table
to identify events occurring simultaneously. Provide
a photocopy of the table and tell students to use three
different coloured highlighters: one colour to highlight
the row with four animals, a second colour to highlight
the row with four supernatural creatures, and a third
colour to highlight vertically the two supernatural
animals. Have students work through the solution.
Refer students to the animation called 1.4 Example 4
Solution for a full explanation of how to determine the
probability of non-mutually exclusive events.
ELL
inclusion, exclusion, gift exchange, ski, cycle
• For visual and tactile learners, provide a photocopy
of the table and coloured counters to represent the
events. Or, have students represent the situation using
Master 1 Venn Diagram.
• Have student pairs develop a similar probability
problem using the table. Encourage them to use a
different method to determine the probabilities. Have
them explain their solution to another student pair.
• For reinforcement of determining the probability of
either of two non-mutually exclusive events, refer
students to www.mathontario.ca Resource Links.
ELL
tokens, role-playing game, dragon, hawk, knight, lion,
princess, witch, wizard, unicorn, supernatural, spells
978-1-25-907746-3
Chapter 1 Introduction to Probability • MHR
29
Assessment Suggestions
Questioning
Assessment as Learning
• Encourage students to describe the concepts from the examples in
their math journal.
• Have students do the Example 1 Your Turn. Check that students
− can determine the probability of a mutually exclusive event using
the sample space
− can determine the probability of a mutually exclusive event by
adding the probabilities of favourable events
• Is the probability of randomly choosing either
an apple juice or a grape juice the same as
or different from the probability of randomly
choosing either a grape juice or an apple juice?
Explain.
• Have students do the Example 2 Your Turn. Check that students
− can calculate the probability of favourable events
− can apply the rule of sum for mutually exclusive events
• Can the rule of sum be used to determine the
probability that Rolly would pick a Connery, a
Dalton, or a Brosnan film from his shelf? Explain.
• Have students do the Example 3 Your Turn. Check that students
− can apply the principle of inclusion and exclusion by adding the
number of family members who like to ski and those who like to
cycle and subtracting the number of family members who like to
ski and cycle
− can determine the probability of randomly choosing someone
who likes to ski or cycle
• Why is the calculated number used to determine
the probability of picking a family member
who likes to ski or cycle different from the total
number of family members who like to ski
added to the total number of family members
who like to cycle?
• Have students do the Example 4 Your Turn. Check that students
− can determine the probability of favourable events
− can calculate the probability of picking either a flying creature or
one that can cast spells
• Have students complete the related parts of BLM 1–1 Chapter 1
Self-Assessment.
• Students may benefit from peer assessment. Have them work in a
small group or in pairs.
• What other method can you use to calculate
the probability that Jozo will randomly choose
a flying creature or one that can cast spells?
Describe how you would use it.
Mathematical Process
Problem Solving
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Question(s)
Reasoning and Proving
Reflecting
Selecting Tools and Computational Strategies
Example 1 Your Turn
Example 3 Your Turn
Connecting
Example 2 Your Turn
Example 4 Your Turn
Representing
Communicating
30 MHR • Chapter 1 Introduction to Probability 978-1-25-907746-3
Consolidate and Debrief
Notes
Meeting Student Needs
As a class, review the Key Concepts. Assign all of
the Reflect questions. Make Master 2 Frayer Model
available for R1. Assign both Practise questions and at
least one of #3 or #4, #5, one of #6 or #7, #8, and at least
one Extend question to students who need a challenge.
Question #9 is an Achievement Check, which can
be used for assessment of learning. For #9c), add a
different range of percents and have students choose
either range to answer the question.
• Have students work in pairs or small groups to
complete the questions. For selected questions, have
a volunteer from each group present the solution to
the class.
• For #9, consider having students work in groups to
complete the question. Ask a volunteer from each
group to present their solution to the class.
• Encourage students to draw a Venn diagram to help
visualize situations. Make Master 1 Venn Diagram
available.
ELL
Frayer model, euchre, take-out, assumptions
Assessment Suggestions
Questioning
Assessment of Learning
• For R1, have students work in pairs to complete the
Frayer model on chart paper. Post the models and have a
volunteer from each group present their work in a math
congress.
• For R2, have student pairs complete a Frayer model using
Master 2 Frayer Model. Then have students write a
response in their math journal. Use the response as an
exit ticket.
• Have students write a response to R3. Have them present
their example to the class.
• What is the probability of randomly drawing a queen or a
jack from a standard deck of cards?
• What is the formula for calculating the probability of two
mutually exclusive events?
• What is the formula for calculating the probability of either
of two non-mutually exclusive events?
• What is the probability of drawing a face card from a
standard deck of playing cards?
• What is the probability of randomly drawing a card in a
standard deck of playing cards that is a 10?
• What is the probability of randomly drawing either a club or
a 10 from a standard deck of cards?
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Learning Goal
Meeting Student Needs
describe how an event
can represent a set of
probability outcomes
• Students may incorrectly calculate the probability of non-mutually exclusive events by
forgetting to subtract the common events. Students who are kinesthetic learners may
benefit from working with manipulatives such as playing cards or marbles to answer the
questions.
• For reinforcement of mutually exclusive and non-mutually exclusive events, go to
www.mathontario.ca Resource Links.
• For extra practice, have students complete BLM 1–10 Section 1.4 Practice.
recognize how different
events are related
• For extra practice, have students complete BLM 1–10 Section 1.4 Practice.
calculate the probability
of an event occurring
• Pair students to discuss solutions.
• A common error is failing to subtract when applying the principle of inclusion and
exclusion. Suggest that students use a Venn diagram or a table to help visualize the
situation.
• For additional practice with the rule of sum and the principle of inclusion and exclusion,
go to www.mathontario.ca Resource Links.
• For extra practice, have students complete BLM 1–10 Section 1.4 Practice.
978-1-25-907746-3
Chapter 1 Introduction to Probability • MHR
31
Mathematical Process
Question(s)
Problem Solving
#8, #12, #14, #15
Reasoning and Proving
#6, #7, #13, #15
Reflecting
R1, #10
Selecting Tools and Computational Strategies
#4, #5
Connecting
R3, #11, #12
Representing
#9
Communicating
R2, #3, #14
Achievement Check Sample Solution
9. a) “e” or a “t”: These are mutually exclusive events.
There is a total of 8 different tiles, with 2 “e”s
and 2 “t”s. The probability of getting an “e” or “t”
2 , or 50%.
2 + __
is __
8 8
red letter or “e”: These are non-mutually exclusive
events. There is a total of 8 different tiles, with 2
“e”s and 3 red letters. The probability of getting a
3 + __
2 – __
1 , or 50%.
red letter or an “e” is __
8 8 8
capital letter or vowel: These are mutually
exclusive events. There is a total of 8 different tiles,
with 1 capital letter and 4 vowels. The probability
4 , or
1 + __
of getting a capital letter or a vowel is __
8 8
62.5%.
not a yellow letter or “t”: These are non-mutually
exclusive events. There is a total of 8 different tiles,
with 2 “t”s and 2 yellow letters. The probability of
2 – __
1 , or 0.375. So, the
2 + __
a “t” or yellow letter is __
8 8 8
probability that she does not choose a yellow or a
“t” is 1 – 0.375, or 62.5%.
b)
u
Red Letters
e
l
J
l
i
t
t
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A
u
red letter or “e”
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J
e
Vowels
J
i
e
e
Capital Letter
u
t
l
t
capital letter or vowel
Not a Yellow
Letter or “t”
J
u
e
Yellow Letters
i
e
i
l
e
t
t
t
e
t
not a yellow letter or “t”
“e” or “t”
32 MHR • Chapter 1 Introduction to Probability 978-1-25-907746-3
c) Answers may vary. Example: What is the
probability that she chooses a blue or black letter?
1.5
Timing
75 min–150 min
Independent and Dependent Events
Materials
• die
• coloured
counters
• bag
• chart paper
• sticky notes
Media Links
• www.mathontario.ca Resource Links:
Compound events, fundamental
counting principle and independent
events, dependent events, conditional
probability
Specific
Expectation
Learning Goal
describe and
determine how one
event occurring can
affect the probability
of another event
occurring
A1.6
solve probability
problems involving
multiple events
A1.6
Blackline Masters
BLM 1–1 Chapter 1 Self-Assessment
BLM 1–3 Chapter 1 Warm-Up
BLM 1–11 Section 1.5 Practice
Sample Success Criteria
•
•
•
•
•
•
•
I can describe a compound event.
I can identify a compound event.
I can describe an independent event.
I can identify an independent event.
I can describe a dependent event.
I can identify a dependent event.
I can recognize the differences between a compound event, an
independent event, and a dependent event.
• I can explain the relationship between dependent and independent
events.
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•
•
•
•
•
•
I can describe the multiplicative principle.
I can recognize when to use the multiplicative principle.
I can describe when to use conditional probability.
I can solve probability problems involving dependent events.
I can solve probability problems involving independent events.
I can apply a variety of strategies to solve probability problems involving
multiple events.
Minds On...
Notes
ELL
Use the photo of six siblings and the questions to
introduce the term compound events. You might survey
students in the class to find out the number and gender
of each student in the class’s siblings and relate the data
to theoretical probability. Challenge students to relate
the probability of a boy or a girl being first born to what
they have learned in their science classes.
gender
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Chapter 1 Introduction to Probability • MHR
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Assessment Suggestions
Questioning
Assessment for Learning
• Use a think-pair-share to address the questions before
discussing as a class.
• Have students complete the section 1.5 questions on
BLM 1–3 Chapter 1 Warm-Up.
• What is the probability that a first child will be a boy? a girl?
• What is the probability that there will be two boys in a
family of two?
• What is the probability that there will be two boys in a
family of two, given that the first child is a boy?
Action!
Example 1 Notes
Meeting Student Needs
Use this example to help students recall terms related
to theoretical probability, such as tree diagram, sample
space, possible outcomes, and favourable outcomes. As
you circulate, check that students fully label their tree
diagram. In a class discussion, challenge students to
think of other ways to represent the sample space for
the Archers. Discuss the answer to part c) as a class and
check that students understand the difference between
multiple events and a single event.
• In their math journal, have students define compound
events and independent events and provide an
example.
• Have students draw a tree diagram for the Your Turn
and explain how to draw each stage.
• Challenge students to draw tree diagrams to represent
the number of outcomes for 4, 5, and 6 children. Ask
them to identify the patterns in the tree diagrams,
and then use the patterns to predict the outcomes for
7 children.
• For reinforcement of finding probability of compound
events, refer students to www.mathontario.ca
Resource Links.
Example 2 Notes
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This example reinforces probability concepts that
students learned in sections 1.1 and 1.2. You may need
to remind students that the notation P(YY) represents
the probability P(Y and Y). Consider presenting the
example as a parallel task and have students develop
a solution for part a) or part b). Encourage volunteers
to show their solution on the board. After, show
students how to calculate the probability of compound
independent events using the multiplicative principle
for independent events. Explain that as the Literacy
Link indicates, it is also called the fundamental
counting principle.
ELL
impact, non-occurrence, influence
Meeting Student Needs
• Encourage students to create a tree diagram. For the
branches, suggest that they use colours that match the
colours of the markers. Do the same for the marbles
in the Your Turn.
• Have students use their math journal to record the
formula and an explanation of the fundamental
counting principle.
• For the Your Turn, remind students that replacing
the marble keeps the sample space the same and
means that the events are independent.
34 MHR • Chapter 1 Introduction to Probability 978-1-25-907746-3
Example 3 Notes
Meeting Student Needs
This example illustrates calculating probability for
different compound events. You may need to remind
students that a composite number can be divided
evenly. Have students work on their own and choose
a method to solve the problem. If they draw a tree
diagram, model how to write the probabilities along the
branches of the tree diagram.
• Before attempting the solution to the example,
consider surveying students to find out if they think
that the game is fair. Have student pairs check their
prediction by playing at least 10 games. They will
need a die, and instead of the spinner they might use
four different coloured counters (including one red)
and draw counters out of a bag.
• For the Your Turn, help students recall the meaning
of a prime number (a number that can be divided
evenly only by 1 or itself).
• For reinforcement of using the fundamental counting
principle, refer students to www.mathontario.ca
Resource Links.
• Challenge students to create a game that includes a
third independent event in addition to the existing
two events in the example. Have them select an
appropriate strategy to determine if the game that
they created is fair.
Example 4 Notes
Meeting Student Needs
This example illustrates calculating probability for
dependent events. Provide student pairs with a bag
and two red counters and two black counters. Have
students compare the probabilities of drawing two red
counters when the first counter drawn is replaced and
when the first counter drawn is not replaced. Explain
that the latter case is an example of dependent events.
Then have students work in small groups and select
a tool or strategy to solve the problem posted in the
example. Have them post their results on chart paper.
Provide time for students to do a gallery walk to see the
solutions and use sticky notes to write comments on
others’ solutions.
• Demonstrate the concept of replacement. Using a bag
that contains four counters, draw a counter and then
put the counter back in the bag. Explain that this is
called “with replacement.” Then demonstrate drawing
a counter and not putting the counter back in the bag.
Explain that this is called without replacement. Ask
students if they think the probabilities for the two
experiments will be different.
• In their math journal, have students write a definition
and an example of dependent events.
• Remind students that not replacing an object reduces
the size of the sample space, and when this happens
the events are dependent. Conversely, replacing an
object keeps the sample size the same and, as a result,
the events are independent.
• Have students work in pairs or in small groups to
complete the Your Turn.
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Chapter 1 Introduction to Probability • MHR
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Example 5 Notes
Meeting Student Needs
This example illustrates conditional probability using
a real-world context. Have students work in groups
to develop a solution using chart paper. Display the
solutions and as a class review the strategies students
used to create the tree diagram.
• Students may benefit from a review of converting
percent to decimals, solving simple formulas, and
calculating the probability of an event not occurring
given the probability of an event occurring.
• Encourage students to record the tree diagram using
words instead of letters for the events.
• For reinforcement, consider revising the number
values for the situation in the example and have
students draw a probability tree diagram to determine
the sales.
• In their math journal have students define and
give an example of conditional probability and the
multiplicative principle for dependent events.
ELL
telemarketing
Assessment Suggestions
Questioning
Assessment as Learning
• Encourage students to describe the concepts from the examples
in their math journal.
• Have students do the Example 1 Your Turn. Check that students
− can create a tree diagram to represent the situation
− can use the tree diagram to calculate the probability of a
simple compound event
• Have students do the Example 2 Your Turn. Check that students
− can create a tree diagram to represent the situation
− can use the tree diagram to calculate the probability of two
independent events occurring
• Have students do the Example 3 Your Turn. Check that students
− can select an appropriate tool and strategy to calculate the
probability of compound events
− can use a tree diagram to calculate the probability of two
independent events occurring
− can determine the probability of different compound events
using the fundamental counting principle
• Have students do the Example 4 Your Turn. Check that students
− can use a tree diagram to calculate the probability of two
dependent events occurring
− can recognize the difference between calculating the
probability of two independent events and the probability of
two dependent events
• Have students do the Example 5 Your Turn. Check that students
− can calculate the probability of an event not happening given
the probability of an event happening
− can create and use a tree diagram to solve a problem involving
conditional probability
• Have students complete the related parts of BLM 1–1 Chapter 1
Self-Assessment.
• Students may benefit from peer assessment. Have them work in a
small group or in pairs.
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• Would the probability of the Singh family having
four girls be the same as or different from the
probability of having a fourth child that is a girl, if
the first three children are girls? Explain.
• Would the probability of randomly drawing a
green marble followed by a yellow marble be the
same or different if the first marble is not replaced
before the second marble is drawn? Explain.
• Would the calculated probability be the same or
different if the die were rolled before spinning the
spinner? Explain.
• How would the probability change if Jelena
replaced the first piece of fruit?
• What is the probability of a randomly chosen
shopper not accepting a sample?
Mathematical Process
Question(s)
Problem Solving
Example 4 Your Turn
Reasoning and Proving
Example 2 Your Turn
Example 3 Your Turn
Reflecting
Selecting Tools and Computational Strategies
Example 5 Your Turn
Connecting
Representing
Communicating
Example 1 Your Turn
Consolidate and Debrief
Notes
Meeting Student Needs
This section allows students to consolidate their
understanding of independent and dependent events
and conditional probability. As a class, review the Key
Concepts. Have students use their math journal to
summarize the key concepts. Assign all of the Reflect
questions. Assign all of the Practise questions and at
least one of #4 or #5, one of #6 or #7, and #8 to #10 of
the Apply questions. Use the Achievement Check for
assessment of learning. Assign the Extend questions to
students who need a challenge.
• Allow students to work in pairs or small groups to
answer the questions.
• For #9, have students use a think-pair-share to solve.
Refer students to the Literacy Link, which explains
how to play the game.
• Remind students to record probabilities along the
branches if they draw a tree diagram.
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maze, contestant, eliminated, overtime, sweeping,
consecutive, going the distance, decision tree
Assessment Suggestions
Questioning
Assessment of Learning
• Set up each of the four
Reflect questions as a
station in the classroom and
have students proceed to
each station.
• For each Reflect question,
have students participate
in a think-pair-share before
writing their response.
• For R1, suppose you draw a card from a deck of cards, replace the card, and then draw
another card. Is this an example of an independent or a dependent event? Explain.
• Suppose you draw a card from a deck of cards and, without replacing the card, draw
another card. What kind of event is this? Explain.
• What strategies can you use to represent and determine the probability for an
independent event? a dependent event?
• For R2, if a coin is flipped three times, does the event that the flip will be a head or a tail
depend on the previous flip?
• If a coin is flipped four times and comes up heads on each toss, does the event that the
flip will be a head on the fifth toss depend on the fourth toss?
• For R3, a player needs to draw two cards of the same suit in order to win and draws a
heart on his first draw. Will the probability that he will draw a heart on his second draw
be conditional on the fact that he has already drawn a heart? Explain.
• For R4, if a telemarketing company finds that out of 500 calls, 20% of people stay on
the line for at least 1 min and the conditional probability of a sale given the previous
condition is 15%, how could a tree diagram be used to determine the number of sales?
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Chapter 1 Introduction to Probability • MHR
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Learning Goal
Meeting Student Needs
describe and determine how the probability of one event
occurring can affect the probability of another event
occurring
• For extra practice, have students complete BLM 1–11
Section 1.5 Practice.
solve probability problems involving multiple events
• For reinforcement of the fundamental counting
principle, dependent events, and conditional probability
go to www.mathontario.ca Resource Links.
• For conditional probability, students may make errors
when determining percent for parts of a tree diagram.
Encourage students to use a highlighter of the same
colour to highlight related conditional probabilities.
• For extra practice, have students complete BLM 1–11
Section 1.5 Practice.
Mathematical Process
Question(s)
Problem Solving
#7, #11
Reasoning and Proving
R2, #5, #9, #10, #13
Reflecting
R4
Selecting Tools and Computational Strategies
#6, #8, #11
Connecting
#12, #14
Representing
#14
Communicating
R1, R3, #4, #9
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Achievement Check Sample Solution
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9. a) There are three possible outcomes for her brother:
rock, paper, and scissors. For Petra to win, her
brother must decide on scissors. The probability
1.
that she wins the car on the first trial is __
3
b) In each round, Petra or Alek play until there are
no (rock-rock) ties. Eliminating the ties leaves
two possible outcomes, each equally likely: rock
smashes scissors (Petra wins) or paper covers rock
(Alek wins); so, the probability that Petra wins
1.
is __
2
38 MHR • Chapter 1 Introduction to Probability 978-1-25-907746-3
c) In part a) there are three possible outcomes,
whereas in part b) there are only two.
d) Assuming random decisions on the parts of the
players, this is a fair game. The tree diagram
for this game shows that there are 9 possible
outcomes: 3 ways for Player A to win, 3 ways for
Player B to win, and 3 ways to tie. The resulting
1,
theoretical probabilities are P(A wins) = __
3
1 , and P(tie) = __
1.
P(B wins) = __
3
3
Review, Test Yourself, and
Chapter Problem Wrap-Up
Timing
75–150 min
Materials
• die
• coloured counters
• bag
• chart paper
• sticky notes
Media Links
• www.mathontario.ca
Resource Links:
Blackjack, Yahtzee,
experimental and
theoretical probability,
conditional probability
Blackline Masters
BLM 1–1 Chapter 1 Self-Assessment
BLM 1–5 Section 1.1 Practice
BLM 1–7 Section 1.2 Practice
BLM 1–8 Section 1.3 Practice
BLM 1–10 Section 1.4 Practice
BLM 1–11 Section 1.5 Practice
BLM 1–12 Chapter 1 Test
BLM 1–13 Chapter 1 Blackline Master Answers
Notes
Meeting Student Needs
Have students complete BLM 1–1 Chapter 1 SelfAssessment and use it to assess their progress and
identify areas they may need to work on. Encourage
students to refer to their math journal notes.
• Have students who require more practice on a topic
refer to BLM 1–5 Section 1.1 Practice, BLM 1–7
Section 1.2 Practice, BLM 1–8 Section 1.3 Practice,
BLM 1–10 Section 1.4 Practice, and BLM 1–11
Section 1.5 Practice.
• For #10, direct students to the Literacy Link that
explains coupe, mini-van, and sedan.
• Have students work together in small groups to
complete the questions.
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quarterback, season, programmed, fleet, snooze
button, alarm, logged on, social media website, pop-up
advertisement, carnival game, dart, servers
Assessment Suggestions
Questioning
Assessment as Learning
• Have students review their math journal notes
and earlier responses on BLM 1–1 Chapter 1 SelfAssessment.
• As students complete the Chapter 1 Review, have them
check their answers in the back of the book. Encourage
students to revisit any sections that they are having
difficulty with prior to starting the Chapter 1 Test
Yourself.
• Where have you seen a similar problem? How did you
solve it?
• How can you show your thinking?
• What tool or strategy would best represent this problem?
• What formula can you use to help solve the problem?
Assessment of Learning
• After students complete the Chapter 1 Test Yourself,
you might want to use BLM 1–12 Chapter 1 Test as a
summative assessment.
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Chapter 1 Introduction to Probability • MHR
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Chapter Problem Wrap-Up Solution
Students report on two or three games in the form of a
written report, electronic slideshow, podcast, poster, or
a format of their choice. Answers may vary.
Example for blackjack:
1. Elements of blackjack that involve strategy include:
• If the dealer turns up a card between 2 and 6,
there is a good chance that the current score in
the dealer’s hand is between 12 and 16, and will
go over 21 if he/she takes an additional card, so
a player should not take an extra card that might
send his score over 21 but rather wait to see what
happens with the dealer’s hand.
• Deciding when to double down: On a double
down, the dealer will give a player one additional
card. The player should take an additional card if
the card total is 9, 10, or 11. The exceptions are
− if the dealer’s card is an ace and the player’s total
is 11
− if the player’s total is 10, the player should double
down unless the dealer has an ace or a 10
− if the player’s total is 9, the player should double
down if the dealer has a 3, 4, 5, or 6
• Deciding when to split pairs: The player can choose
to split pairs, which means separating two cards
into two hands.
− always split aces and 8s
− do not split 5s and 10s
2. Elements of blackjack that involve chance or
probability include:
• There is a 30% chance that the dealer has a
“blackjack” (a score of 21).
• The player should double down, which means
taking an additional card, if the card total is 9,
10, or 11, since there is a good chance that a) the
additional card will result in a hard score for the
dealer to beat or b) if the additional card is a low
scoring card, there is still a chance that the dealer
will go over 21.
3. There is a relative balance of strategy versus chance
in blackjack. The strategy is for the player to achieve
a hand with a points total closer to 21 than the dealer
without going over 21. If a player uses strategy and
counting cards, the player has a fair chance against
the house, but each card that is dealt in each game is
up to chance. The dealer has an advantage over the
player in that the dealer plays last and will win when
a player breaks 21, even if the dealer breaks 21 in the
same round.
• Three outcomes unique to the game:
− Since there are 52 cards and 4 cards in each suit,
the probability of drawing a particular card, such
1 ≈ 8%.
4 = ___
as 5: P(5) = ___
52 13
− The most likely possibility is playing a 10, since
4 out of 13 cards in each suit have a value of 10.
3 ≈ 23%.
12 = ___
P(10) = ___
52 13
− If the dealer’s hole card (card hidden from
view) has a value of 10, the dealer will lose if he
draws any card greater than 5, which is 32 out of
32
52 cards: P(card drawn is greater than a 5) = ___
52
8 ≈ 61.5%.
= ___
13
For information and blackjack rules, go to
www.mathontario.ca Resource Links.
Example for Yahtzee:
1. Elements of Yahtzee that involve strategy include:
• Getting the highest number of points possible.
− A player should always go for the Yahtzee. The
first Yahtzee is worth 50 points and additional
Yahtzees are worth 100 points each.
• Getting bonus points. There are two sections
on the score card. If the total of all points in the
upper section is 63 or greater, there is a bonus of
35 points. Players should keep this in mind when
deciding which slot to fill in for a roll that can go in
more than one place.
− Fill in the upper section of the score card with
high scores.
− For a roll of a kind with four 4s, 5s, or 6s, record
the points in the upper section of the score card,
not in the four of a kind box.
• Fill up the most valuable slots early so that the
player will be out fewer points if 0s need to be
taken at the end of the game.
• A player doesn’t have to choose the box that gives
the highest combination for the score that has
been rolled. It might be better to save that slot for a
better roll later.
2. The element of Yahtzee that involves chance is rolling
the dice.
3. There is a relative balance of strategy versus chance
in the game of Yahtzee. The strategy of the game is
to get the highest number of points and get enough
points in the upper level of the score card to score a
bonus, but the roll of the dice is up to chance.
• Three outcomes unique to the game:
− The probability of getting a Yahtzee in a single
1
roll: After the first die is rolled, there is a __
6
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probability that the second die is the same as
1 probability that the third die is
the first die, a __
6
1 probability that the
the same as the first die, a __
6
fourth die is the same as the first die, and a
1 probability that the fifth die is the same as the
__
6
first die. Therefore, the probability of a Yahtzee
being rolled in one roll is
1 × __
1 × __
1 × __
1 = _____
1
1 × __
P(Yahtzee) = __
1 6
6
6
6 1296
≈ 0.08%.
Test Yourself Question
(Achievement Chart
Category)
Section
− The probability of rolling a pair from five dice:
The sample space is 6 × 6 × 6 × 6 × 6 = 7776. The
favourable outcomes (two dice are the same) =
4680. The probability of rolling a pair is P(rolling
65 ≈ 60.19%.
4680 = ____
a pair) = _____
7776 108
− The probability of rolling four of a kind is
25 ≈ 1.93%.
P(rolling four of a kind) = _____
1296
For information and rules, go to www.mathontario.ca
Resource Links.
Learning Goal
Meeting Student Needs
• Review how to calculate theoretical
probability
• Remind students to draw a tree diagram
• Review techniques of handling multiple
choice questions by eliminating the
distractors
1
(Knowledge/Understanding)
1.2
• calculate theoretical
probability
2
(Knowledge/Understanding)
1.1
• measure and calculate
simple probabilities
• recognize the difference
between experimental
probability and theoretical
probability
• Review how to calculate experimental
probability. Remind students to divide
the number of favourable outcomes by
the total number of trials
1.3
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(Knowledge/Understanding)
1.3
• recognize the difference
between experimental
probability and theoretical
probability
• For reinforcement of experimental
and theoretical probability, go to
www.mathontario.ca Resource Links
4
(Application)
1.2
• calculate theoretical
probability
• For reinforcement of calculating
probability, go to www.mathontario.ca
Resource Links
5
(Communication)
1.2
• calculate theoretical
probability
• Review subjective probability and odds
in favour
• Refer students to 1.2 Example 3 and their
math journal entries
6
(Thinking)
1.2
• use probability to
describe the likelihood of
something occurring
• measure and calculate
simple probabilities
• Have student pairs use the diagram to
create a similar probability problem for
practice. Have them exchange and solve
each other’s problem
• Refer students to 1.2 Example 2
7
(Communication)
1.2
• calculate theoretical
probability
• Review odds against and odds in favour
8
(Application)
1.4
• calculate the probability of
an event occurring
• Review the rule of sum
• Refer students to 1.4 Example 2
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Test Yourself Question
(Achievement Chart
Category)
Section
Learning Goal
Meeting Student Needs
9
(Application)
1.5
• solve probability involving
multiple events
• Have students identify key words in the
problem
• Refer students to 1.5 Example 4
• Remind students to draw a tree diagram
10
(Thinking)
1.5
• solve probability problems
involving multiple events
• Refer students to 1.5 Example 5
• For reinforcement of calculating
conditional probability, go to
www.mathontario.ca Resource Links
• Challenge students to create web links
about probability that might be helpful
for their classmates
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