11.1
Compound Events
Lesson Objectives
• Understand compound events.
• Represent compound events.
Vocabulary
compound event
simple event
possibility diagram
tree diagram
Understand Compound Events.
A compound event consists of two or more simple events occurring together or
one after another. For example, tossing a coin for heads or rolling a 3 on a six-sided
number die are both simple events. But tossing a coin for heads and rolling a 3 on a
six-sided number die is a compound event.
Example 1
Identify events as simple or compound.
Tell whether each event is a simple or compound event. State the single event or
identify the simple events that form the compound event.
a)
Getting a number less than 2 or greater than 4 when spinning the spinner once
1
Solution
This is a simple event. There is one event: getting a number less than 2 or greater
than 4 from one spin of the spinner.
b)
2
8
3
7
4
6
5
Getting a number less than 2 or greater than 4 when spinning the spinner two
times consecutively
Solution
This is a compound event. There are two simple events: getting a number less than 2
or greater than 4 one after another.
c)
Getting heads when a coin is tossed and getting a 3 when a six-sided number
die is rolled
Solution
This is a compound event. There are two simple events: getting heads on a coin and
a 3 on a six-sided number die.
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Guided Practice
Tell whether each outcome is from a simple or compound event. If it is a
compound event, identify the simple events that form the compound event.
1 Obtaining two heads when two coins are tossed
2 Winning a football game
3 Getting a number less than 4 or greater than 5 when a fair six-sided number
die is rolled
4 Rolling two fair six-sided number dice and obtaining a sum of 10 from the throws
Represent Compound Events.
Suppose you roll a fair six-sided number die and toss a fair coin.
The simple events that form this compound event are rolling a number on the
number die and tossing the coin for heads and tails.
There are six possible outcomes when a number die is rolled. The sample
space is {1, 2, 3, 4, 5, 6}.
There are two possible outcomes when a coin is tossed. They are {H, T}, where H
denotes the outcome Heads, and T denotes the outcome Tails.
The braces { } are used
to list the set of possible
outcomes in a sample
space or outcomes
favorable to an event.
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There are many ways to represent and display all the outcomes of a compound
event. An organized list for the outcomes of tossing a number die and a coin is
shown here.
Die
1
1
2
2
3
3
4
4
5
5
6
6
Coin
H
T
H
T
H
T
H
T
H
T
H
T
Outcome
1H
1T
2H
2T
3H
3T
4H
4T
5H
5T
6H
6T
A two-way grid or a table is a type of possibility diagram that can help you visualize
all the possible outcomes of a compound event. You can also circle or mark the
favorable outcomes.
List the outcomes for rolling a die on the horizontal axis and the outcomes for tossing
a coin on the vertical axis. Note that each intersection of grid lines represents a
possible outcome of the compound event.
T
Coin
Tossed
H
1
2
3
4
5
6
H represents heads
T represents tails
Number Die Rolled
Think Math
How can the possibility diagram
help you determine the number
of possible outcomes in a sample
space of a compound event without
counting?
Because the two simple
events occur together, the
order of the events is not
important. So, you can use
a dot (•) to indicate each
possible outcome.
From the diagram, you can see that there are 12 possible outcomes in the sample
space for this compound event. Notice that the outcomes in the two-way grid are the
same as those in the organized list for this sample space.
Continue on next page
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Another diagram that is used to display the outcomes of a compound event is a table
of ordered pairs.
1
2
1
2
3
3
Spinner 1
Spinner 2
Suppose there are two spinners, with each spinner divided into three equally-sized
angles at the center.
The row labels and column labels of the table list the outcomes of each simple event.
Each possible outcome is written in the diagram as an ordered pair:
(first event, second event). You can see that there are 3 · 3 5 9 possible outcomes in
the sample space.
Spinner 2
Spinner 1
1
2
3
1
(1, 1)
(2, 1)
(3, 1)
2
(1, 2)
(2, 2)
(3, 2)
3
(1, 3)
(2, 3)
(3, 3)
Example 2
Represent all possible outcomes of a compound event.
Represent and tell the number of possible outcomes for each compound event
described.
a) The results of rolling two fair six-sided number dice are added.
Solution
2nd Toss
1st Toss
Math Note
+
1
2
3
4
5
6
1
2
3
4
5
6
7
2
3
4
5
6
7
8
3
4
5
6
7
8
9
4
5
6
7
8
9
10
5
6
7
8
9
10
11
6
7
8
9
10
11
12
You can write the operation of the
compound event at the top left cell
of the table to indicate that you are
finding the sum of the outcomes of
two events.
There are 36 possible outcomes.
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b) The two spinners shown below are spun.
1
0
2
1
2
4
3
Spinner 1
Spinner 2
Solution
Spinner 2
Spinner 1
0
1
2
4
1
(0, 1)
(1, 1)
(2, 1)
(4, 1)
2
(0, 2)
(1, 2)
(2, 2)
(4, 2)
3
(0, 3)
(1, 3)
(2, 3)
(4, 3)
There are 12 possible outcomes.
c)One drawer has four shirts: 1 blue, 1 yellow, 1 red, and 1 gray. Another
drawer has two pairs of socks: 1 gray and 1 black. A shirt and a pair of
socks are taken from its drawer.
Solution
Socks
Gray
Black
Blue Yellow Red Gray
Shirts
There are 8 possible outcomes.
Guided Practice
Represent and tell the number of possible outcomes for each compound event
described.
5 The results of tossing two fair coins together
6 The results of rolling two fair six-sided number dice are multiplied.
7 A fair six-sided number die and a fair four-sided number die labeled
1 to 4 are rolled. The results that face down on both number
dice are recorded.
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Represent Compound Events Using Tree Diagrams.
A tree diagram is another type of possibility diagram that can be used to represent a
compound event. The tree diagram below represents the outcomes of a simple event,
tossing a fair coin. The branches from the node represent all possible outcomes.
1
2
H
Branches
Node
1
2
H represents heads
T represents tails
T
When all the branches represent equally
likely outcomes, you can omit labeling
the probabilities on the branches.
For drawing any tree diagram, you should take note of the following:
• Each branch starts from the same node.
• The number of branches indicates the number of outcomes the event has.
• The outcome for the event is written at the end of a branch.
• The probability of the outcome of an event is written in parentheses along
the branch.
• The probabilities of the branches from each node must add up to 1.
If the coin is tossed twice, one after another, the tree diagram looks like this:
1st Toss
2nd Toss
Outcome
H
(H, H)
H
T
(H, T)
H
(T, H)
T
T
(T, T)
Math Note
Listing the outcomes in a column is
optional in a tree diagram. You can also
determine the number of outcomes,
when all equally likely outcomes are
shown, by counting the number of
branches at the last event in the tree.
H represents heads
T represents tails
You can see from the tree diagram that there are 4 equally likely possible outcomes.
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Example 3
a)
Represent a compound event using a tree diagram.
Robyn has a fair spinner and a coin as shown. She first spins
the spinner once and then tosses the coin. Draw a tree
diagram to represent all possible outcomes. Then tell
the number of possible outcomes.
R
B
G
Solution
First, draw branches for each outcome
of the first event, the spinner. The end
of each branch becomes a node for
the second event, tossing a coin.
Spinner
Coin
Outcome
H
(R, H)
T
(R, T)
H
(B, H)
T
(B, T)
R
B
H
(G, H)
T
(G, T)
G
R represents red
B represents blue
G represents green
H represents heads
T represents tails
There are 6 possible outcomes in this compound event.
Think Math
How would you draw the tree
diagram if the two simple events are
switched: first, toss heads or tails on
the coin, and second, spin a color
on the spinner?
Continue on next page
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b)
Eric has a yellow, a pink, and a green highlighter in his pencil case. He also has
1 red pen and 2 black pens. Eric randomly selects a highlighter and a pen. Draw
a tree diagram to represent all possible outcomes. Then tell the number of
possible outcomes.
Solution
Highlighter
Y
P
G
Pen
Outcome
R
(Y, R)
B
(Y, B)
B
(Y, B)
R
(P, R)
B
(P, B)
B
(P, B)
R
(G, R)
B
(G, B)
B
(G, B)
Y represents yellow
P represents pink
G represents green
R represents red
B represents black
There are 9 possible outcomes in this compound event.
Think Math
Are the outcomes shown at the end
of the branches equally likely to
occur? Explain.
Guided Practice
For each compound event, draw a tree diagram to represent the possible
outcomes. Then tell the number of possible outcomes.
8 Joshua has two bags. The first bag contains 2 blue beads and 1 green bead.
The second bag contains 3 lettered cards with the letters P, Q, and R. Joshua
randomly takes an item from the first bag, and then from the second bag.
9 A fair coin is tossed, and then a fair four-sided color die with faces painted yellow,
green, blue, and black is rolled. The color facing down is the result recorded.
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Practice 11.1
Tell whether each statement is True or False.
1 Selecting the letter A from the word PROBABILITY is a compound event.
2 Selecting the letter B from the word BASEBALL and then from the word ABLE is a
simple event.
3 Tossing a fair six-sided number die to get either an even number or a five is a
compound event.
4 Umberto has 3 red cards and 4 blue cards. Drawing two red cards in a row, without
replacing the first card before drawing the second card, is a compound event.
Tell whether each event is a simple or compound event. If it is a compound
event, identify the simple events that form the compound event.
5 Getting a 6 when a fair six-sided number die is rolled.
6 Rolling three fair six-sided number dice and obtaining a sum of 18 from the throws.
7 Getting an eighteen when a fair twenty-sided number die is rolled.
8 Susan has 3 red cards and 4 blue cards. She first draws a blue card. Without
replacing the first card, she then draws another blue card.
Solve. Show your work.
9 In the top drawer, there are two battery-operated flashlights: red and yellow. In the
second drawer, there are three packages of batteries: sizes AA, C, and D. A flashlight
and a package of batteries are randomly selected.
a)
Draw a possibility diagram to represent all possible outcomes.
b)
How many possible outcomes are there?
10 Two electronic spinners, A and B, are spun by pressing a button. Spinner A has four
sections labeled 1 to 4, while B has three sections, labeled 1 to 3. Spinner B, due to a
technical error, will never land on number 2 if spinner A lands on a 4.
a)
Draw a possibility diagram to represent all possible outcomes.
b)
How many possible outcomes are there?
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11 Winston has two boxes. The first box has 3 black pens and 1 red pen. The second
box has 1 green ball and 1 yellow ball. Draw a tree diagram to represent all
possible outcomes for randomly drawing a pen and a ball from each box. Then
tell the number of possible outcomes.
12 Seraphina first tosses a fair six-sided number die. She then tosses a fair coin. Draw
a tree diagram to represent all possible outcomes.
13 A game was designed such that a participant needs to accomplish 2 rounds to be
considered the overall winner. The first round is to roll a 4 from a fair four-sided
number die labeled 1 to 4. The result recorded is the number facing down. The
second round is to randomly draw a red ball from a box of 2 differently colored balls.
a) Draw a tree diagram to represent all possible outcomes.
b) How many possible outcomes are there?
c)
If the participant first draws the colored ball and then
rolls the four-sided number die, will the number of possible outcomes
be the same? Draw a tree diagram to explain your reasoning.
14 Zoe first rolls a fair four-sided number die labeled 1 to 4. Then she rolls another
fair four-sided number die labeled 2 to 5. The results recorded are the numbers
facing down.
a) Draw a possibility diagram to find the number of favorable outcomes for an
odd sum.
b) Draw a possibility diagram to find the number of favorable outcomes for a
difference greater than 2.
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