1. No category

Name Class 8-6
Date Binomial Distributions
Extension: Probability Distributions
Essential question: What is a probability distribution and how is it displayed?
1
CC.9–12.S.MD.5(+)
ENGAGE
Video Tutor
Introducing Probability Distributions
A probability distribution is a data distribution that gives the
probabilities of the values of a random variable. A probability
distribution can be represented by a histogram in which the values
of the random variable—that is, the possible outcomes—are on the
horizontal axis, and probabilities are on the vertical axis. The figure
shows the probability distribution for rolling a number cube.
Probability
A random variable is a variable whose value is determined by the outcome of a
probability experiment. For example, when you roll a number cube, you can use a
random variable X to represent the number you roll. The possible values of X are
1, 2, 3, 4, 5, and 6.
When the values of a random variable are consecutive whole
numbers, as is the case for rolling a number cube, a histogram for the
probability distribution typically shows bars that each have a width
of 1 and is centered on a value of the variable. The area of each bar
therefore equals the probability of the corresponding outcome, and the
combined areas of the bars is the sum of the probabilities, which is 1.
3
6
2
6
1
6
1
2
3
4
5
6
Result of rolling
number cube
REFLECT
1a. In an experiment in which a coin is tossed twice, the random variable X is the
number of times that the coin lands heads up. What are the possible values of the
random variable?
1b. A spinner has 8 equal sections, each labeled 1, 2, 3, or 4. The
histogram shows the probability distribution for spinning the
spinner. How many sections of the spinner are labeled with each
number? How do you know?
1
2
Probability
© Houghton Mifflin Harcourt Publishing Company
A cumulative probability is the probability that a random variable is less than or equal to
a given value. You can find cumulative probabilities from a histogram by adding the areas
of the bars for all outcomes less than or equal to the given value.
3
8
1
4
1
8
1
2
3
4
Result of spinning
a spinner
Chapter 8
473
Lesson 6
CC.9–12.S.MD.5(+)
2
EXAMPLE
Displaying a Probability Distribution
You roll two number cubes at the same time. Let X be a random variable that
represents the sum of the numbers rolled. Make a histogram to show the
probability distribution for X.
A Complete the frequency table to show the number of ways that you can get each sum
in one roll of the number cubes.
Sum
2
Frequency
1
3
4
5
6
7
8
9
10
11
12
B Add the frequencies you found in part A to find the total number of possible outcomes.
The total number of possible outcomes is
.
C Divide each frequency by the total number of outcomes to find the probability of each
sum. Complete the table.
Sum
2
Probability
___
​  1  ​ 
3
4
5
6
7
8
9
10
11
12
36
D Create a histogram with the sums on the horizontal axis and the probabilities on the
vertical axis. Complete the histogram below by labeling the axes and drawing a bar to
represent the probability of each sum.
Probability
© Houghton Mifflin Harcourt Publishing Company
1
36
1
Sum
REFLECT
2a. The probability that you roll a sum less than or equal to 5 is written P(X ≤ 5). What
is this probability? How is it represented in the histogram?
Chapter 8
474
Lesson 6
In the example, you used theoretical probabilities to define a probability distribution.
You can also use experimental probabilities to define a probability distribution.
CC.9–12.S.IC.2
3
EXPLORE
Using a Simulation
You flip a coin 7 times in a row. Use a simulation to determine the probability
distribution for the number of times the coin lands heads up.
A When you flip a coin, the possible outcomes are heads and tails. You will use your
calculator to generate random numbers between 0 and 1, assigning heads to numbers
less than or equal to 0.5 and tails to numbers greater than 0.5.
and then select PRB. Choose 1:rand and
To do the simulation, press
press ENTER .
Now press ENTER 7 times to generate 7 random numbers. This
simulates one trial (that is, one set of 7 coin flips). Record the
number of heads in the table. For example, on the calculator
screen shown here, there are 3 numbers less than or equal to 0.5,
so there are 3 heads.
Carry out three more trials and record your results in the table.
Trial
1
MATH
2
3
4
Number of Heads
B Report your results to your teacher in order to combine everyone’s results. Use the
combined class data to complete the table below. To find the relative frequency for an
outcome, divide the frequency of the outcome by the total number of trials in the class.
© Houghton Mifflin Harcourt Publishing Company
Number of Heads
0
1
2
3
4
5
6
7
Frequency
Relative
Frequency
C Enter the outcomes (0 through 7) into your calculator as list ​L​1​. Enter the relative
frequencies as list ​L2​ ​.
D Make a histogram by turning on a statistics plot, selecting the histogram option, and
using ​L1​ ​for Xlist and ​L​2​for Freq. Set the viewing window as shown. Then press
A sample histogram is shown below.
Chapter 8
475
GRAPH
.
Lesson 6
REFLECT
3a. Describe the shape of the probability distribution.
3b. Based on the histogram, what is P(X ≤ 3)? That is, what is the probability of getting
3 or fewer heads when you flip a coin 7 times? Explain.
3c. If you flipped a coin 7 times and got 7 heads, would this cause you to question
whether the coin is fair? Why or why not?
CC.9–12.S.MD.3(+)
4
example
Analyzing a Probability Distribution
The histogram shows the theoretical probability distribution
for the situation in the Explore. Use the distribution to answer
each question.
P(X ≥ 4) = P(X = 4) + P(X = 5) + P(X = 6) + P(X = 7)
So, the probability of getting 4 or more heads is
+
+
=
.
0.273 0.273
0.2
0.164
0.1
0.055
An easy way to calculate this probability is to use the
complement of the event. The complement of getting
at least 1 head is getting 0 heads. Use the histogram
to find P(X = 0) and subtract it from 1.
P(X = 0) =
So, the probability of getting at least 1 head is
1-
=
0.055
0.008
0.008
B What is the probability of getting at least 1 head?
0.164
0 1 2 3 4 5 6 7
Number of Heads
.
REFLECT
4a. Why are the probabilities in the histogram you made in the Explore different from
the probabilities given in the histogram above?
Chapter 8
476
Lesson 6
© Houghton Mifflin Harcourt Publishing Company
+
Probability
A What is the probability of getting 4 or more heads?
0.3
4b. What do you think would happen to the histogram you made in the Explore if you
included data from 1000 additional trials?
4c. Why does it make sense that the histogram that shows the theoretical probabilities
is symmetric?
pra c t i c e
1. The spinner at right has three equal sections. You spin the spinner twice and
find the sum of the two numbers the spinner lands on.
a. Let X be a random variable that represents the sum of the two numbers.
What are the possible values of X?
2
1
3
b. Complete the table.
© Houghton Mifflin Harcourt Publishing Company
Sum
Probability
c. Make a histogram of the probability distribution.
1
d. What is the probability that the sum is not 2? How is this probability represented
in the histogram?
Chapter 8
477
Lesson 6
2. You roll two number cubes at the same time. Let X be a random variable that
represents the absolute value of the difference of the numbers rolled.
a. What are the possible values of X?
b. Complete the table.
Difference
Probability
c. Is this probability distribution symmetric? Why or why not?
3. What is the probability of getting 6 or 7 heads?
0.4
Probability
A trick coin is designed to land heads up with a probability
of 80%. You flip the coin 7 times. The histogram shows the
probability distribution for the number of times the coin
lands heads up. (“0+” means slightly greater than 0.) Use
the histogram for Exercises 3-6.
0.367
0.3
0.275
0.004
0.1
0.210
0.029
0.2
0.115
0+
0+
0 1 2 3 4 5 6 7
4. What is the probability of getting 4 or more heads? Explain.
Number of Heads
© Houghton Mifflin Harcourt Publishing Company
5. Is the probability of getting an even number of heads the same as the probability of
getting an odd number of heads? Explain.
6. Suppose you flip a coin 7 times and get 7 heads. Based on what you know now,
would you question whether the coin is fair? Why or why not?
Chapter 8
478
Lesson 6
8-6
Name Class Date __________________
Date Class __________________
Name
________________________________________
Practice
Additional
Practice
2
Chapter
1. You roll two four-sided number cubes at the same time. Let X be a random variable that
represents the product of the numbers rolled.
a. What are the possible values of X?
_________________________________________________________________________________________
b. Complete the table.
Control
Probability
c. Draw a histogram of the probability distribution.
2. Describe the probability distribution for flipping a fair coin. Does this depend on the
number of flips?
© Houghton Mifflin Harcourt Publishing Company
_________________________________________________________________________________________
_________________________________________________________________________________________
_________________________________________________________________________________________
Original
content
the original content are the responsibility of the instructor.
Chapter
8 Copyright © by Holt McDougal. Additions and changes to479
97
1
Lesson 6
Holt McDougal Algebra 1
Name ________________________________________ Date __________________ Class __________________
Problem
Solving
Problem
Solving
Chapter
Binomial Distributions
2
Sales records for the snack machines show that 1 of every 6 students
buys a bag of trail mix. There are 5 students waiting to use the
machines. Melanie uses the formula for binomial probability,
−
P(r) = nCrprqn r, to determine the number of students expected to buy
trail mix. (The expression nCr means
n!
).
( r ! )( n − r ! )
1. What is the probability of exactly 3 students buying a bag of trail mix?
a. What is the probability of each student buying a
bag of trail mix?
b. Define each variable used in the formula and give its value.
c. Write the binomial formula, substituting these values. __________________________
d. Solve the equation to give the probability of exactly
3 students buying a bag of trail mix.
2. Repeat the process to find the probability of exactly 0, 1, 2, 4, and 5 students buying a bag
of trail mix. Use these results to graph a probability distribution.
© Houghton Mifflin Harcourt Publishing Company
3. What is the probability of at least 1 student buying a bag of trail mix?
a. Describe a method to solve involving the sum of probabilities.
b. Describe a method to solve that uses the formula P(E) +P(not E) = 1.
c. Use either method to determine the probability of at least
1 student buying a bag of trail mix.
______________________
Original content Copyright © by Holt McDougal. Additions and changes to the original content are the responsibility of the instructor.
Chapter 8
480
98
1
Lesson 6
Holt McDougal Algebra 1