Lesson 10
NYS COMMON CORE MATHEMATICS CURRICULUM
M5
PRECALCULUS AND ADVANCED TOPICS
Lesson 10: Determining Discrete Probability Distributions
Student Outcomes

Given a description of a discrete random variable, students determine the probability distribution of that
variable.

Students interpret probabilities in context.
Lesson Notes
In this lesson, students are again given a description of a chance experiment that results in a discrete random variable.
Students derive the discrete probability distribution for that random variable and use it to answer probability questions
and interpret those probabilities in context. As in the previous lesson, students mathematically determine probability
distributions based on the possible outcomes of an event. In this lesson, they realize that only after many trials do
distributions of event outcomes approach those in the calculated probability distribution. Each student needs two
pennies for this lesson.
Classwork
Exercise 1 (2 minutes)
Have students read and answer Exercise 1 independently. This should serve as a refresher from the previous lesson, but
it also provides the necessary probability distribution for the forthcoming experiment. Discuss the answer when the
class is finished. Make sure that students carefully explain their reasoning.
Scaffolding:
Exercise 1
Recall this example from Lesson 9:
A chance experiment consists of flipping a penny and a nickel at the same time. Consider the
random variable of the number of heads observed.
The probability distribution for the number of heads observed is as follows:
Number of Heads
Probability
1.
𝟎
𝟏
𝟐
𝟎. 𝟐𝟓
𝟎. 𝟓𝟎
𝟎. 𝟐𝟓
What is the probability of observing exactly 𝟏 head when flipping a penny and a nickel?
The probability of observing exactly 𝟏 head when flipping a penny and a nickel is 𝟎. 𝟓𝟎.
 The teacher could
demonstrate the
experiment by flipping a
penny and a nickel at the
same time and finding some
probabilities empirically
first in a chart similar to the
one shown.
 More advanced students
could answer a more
challenging question such
as a similar question
involving three coins.
Exercises 2–7 (12 minutes)
In these exercises, students flip two coins a small number of times and create an actual probability distribution. Have
students complete Exercises 2–7 independently. Discuss the answers once the class has finished. As students share
their probability distributions from Exercise 6, the distributions are expected to vary.
Lesson 10:
Determining Discrete Probability Distributions
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Lesson 10
NYS COMMON CORE MATHEMATICS CURRICULUM
M5
PRECALCULUS AND ADVANCED TOPICS
Exercises 2–7
2.
Suppose you will flip two pennies instead of flipping a penny and a nickel. How will the probability distribution for
the number of heads observed change?
The probability distribution for the number of heads observed when flipping two pennies would be the same as the
probability distribution for the number of heads observed when flipping a penny and a nickel. This is because both
the penny and the nickel have a 𝟓𝟎% chance of landing on heads.
3.
Flip two pennies, and record the number of heads observed. Repeat this chance experiment three more times for a
total of four flips.
Student answers will vary. One example is shown below.
4.
Flip
Number of
Heads
𝟏
𝟎
𝟐
𝟎
𝟑
𝟏
𝟒
𝟏
What proportion of the four flips resulted in exactly 𝟏 head?
Student answers will vary. Based on the sample answer in Exercise 3, the proportion of the four flips in which I
observed exactly 𝟏 head is 𝟎. 𝟓𝟎.
5.
Is the proportion of the time you observed exactly 𝟏 head in Exercise 4 the same as the probability of observing
exactly 𝟏 head when two coins are flipped (given in Exercise 1)?
Student answers will vary. Based on the sample answer in Exercise 3, the proportion in Exercise 4 is the same as the
probability in Exercise 1. However, this will not be the case for all students.
6.
Is the distribution of the number of heads observed in Exercise 3 the same as the actual probability distribution of
the number of heads observed when two coins are flipped?
Student answers will vary. Based on the sample answer in Exercise 3, my distribution for the number of heads
observed will not be the same. One answer is given below.
Number of Heads
Proportion
7.
𝟎
𝟏
𝟐
𝟎. 𝟓𝟎
𝟎. 𝟓𝟎
𝟎
In Exercise 6, some students may have answered, “Yes, they are the same.” But many may have said, “No, they are
different.” Why might the distributions be different?
Since the chance experiment of flipping two pennies is only repeated four times, the proportions for the number of
heads observed will probably not be the same as the actual probabilities for the number of heads observed.
Lesson 10:
Determining Discrete Probability Distributions
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Lesson 10
NYS COMMON CORE MATHEMATICS CURRICULUM
M5
PRECALCULUS AND ADVANCED TOPICS
Exercises 8–9 (10 minutes)
In the previous exercises, students flipped two coins four times and recorded the distributions of the outcomes. Because
the number of trials was so small, there was probably a wide range of results. In this exercise, when all the trials are
combined as a class, the overall distribution should be closer to the calculated distribution. Put the following table up on
the board, and have students make tally marks in the appropriate cells to add their four observations from Exercise 3.
Number of Heads
0
1
2
Tally
Then, have students calculate the proportions needed for Exercise 8. After students have completed Exercise 9, remind
them that the probabilities describe the long-run behavior of the random variable number of heads observed when two
pennies are flipped, and introduce the law of large numbers.

In the case of coins, the more times the two pennies are flipped, the observed probabilities of getting 0, 1, and
2 heads should be getting closer to what values?


In the case of coins, the more times two pennies are flipped, the closer the observed probabilities of 0,
1, and 2 heads get to the calculated distribution of 0.25, 0.50, and 0.25, respectively.
Even though students get varying results flipping on their own, when combined as a class, the results should
start to approach the calculated distribution.
Exercises 8–9
Number of Heads
𝟎
𝟏
𝟐
Tally
8.
Combine your four observations from Exercise 3 with those of the rest of the class on the chart on the board.
Complete the table below.
Class answers will vary. One example is given.
Number of Heads
Proportion
9.
𝟎
𝟏
𝟐
𝟎. 𝟐𝟖
𝟎. 𝟒𝟕
𝟎. 𝟐𝟓
How well does the distribution in Exercise 8 estimate the actual probability distribution for the random variable
number of heads observed when flipping two coins?
The proportions for the number of heads observed are approximately equal to the actual probabilities for the
number of heads observed.
The probability of a possible value is the long-run proportion of the time that value will occur. In the above scenario, after
flipping two coins many times, the proportion of the time each possible number of heads is observed will be close to the
probabilities in the probability distribution. This is an application of the law of large numbers, one of the fundamental
concepts of statistics. The law says that the more times an event occurs, the closer the experimental outcomes naturally
get to the theoretical outcomes.
Lesson 10:
Determining Discrete Probability Distributions
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Lesson 10
NYS COMMON CORE MATHEMATICS CURRICULUM
M5
PRECALCULUS AND ADVANCED TOPICS
Exercises 10–12 (13 minutes)
Scaffolding:
Give students time to read the text, and then have them complete Exercise 10
independently. Discuss the answer when the class is finished. When there is agreement
on the answer to Exercise 10, have students complete Exercises 11–12. Discuss the
answers when the class is finished.
Exercises 10–12
A May 2000 Gallup poll found that 𝟑𝟖% of the people in a random sample of 𝟏, 𝟎𝟏𝟐 adult
Americans said that they believe in ghosts. Suppose that three adults will be randomly selected
with replacement from the group that responded to this poll, and the number of adults (out of
the three) who believe in ghosts will be observed.
 Alternatively, read the text
aloud, and ask students to
restate the problem in their
own words to a partner.
 It might also help to explain
a little about the word
Gallup: Gallup is a worldfamous polling organization
known for its public opinion
polls. It was founded by
George Gallup in 1935.
10. Develop a discrete probability distribution for the number of adults in the sample who
believe in ghosts.
MP.4
Person 1
Person 2
Person 3
Calculation
Probability
NG
NG
NG
𝟎. 𝟔𝟐 × 𝟎. 𝟔𝟐 × 𝟎. 𝟔𝟐
𝟎. 𝟐𝟑𝟖 𝟑𝟐𝟖
NG
NG
G
𝟎. 𝟔𝟐 × 𝟎. 𝟔𝟐 × 𝟎. 𝟑𝟖
𝟎. 𝟏𝟒𝟔 𝟎𝟕𝟐
NG
G
NG
𝟎. 𝟔𝟐 × 𝟎. 𝟑𝟖 × 𝟎. 𝟔𝟐
𝟎. 𝟏𝟒𝟔 𝟎𝟕𝟐
NG
G
G
𝟎. 𝟔𝟐 × 𝟎. 𝟑𝟖 × 𝟎. 𝟑𝟖
𝟎. 𝟎𝟖𝟗 𝟓𝟐𝟖
G
NG
NG
𝟎. 𝟑𝟖 × 𝟎. 𝟔𝟐 × 𝟎. 𝟔𝟐
𝟎. 𝟏𝟒𝟔 𝟎𝟕𝟐
G
G
NG
𝟎. 𝟑𝟖 × 𝟎. 𝟑𝟖 × 𝟎. 𝟔𝟐
𝟎. 𝟎𝟖𝟗 𝟓𝟐𝟖
G
NG
G
𝟎. 𝟑𝟖 × 𝟎. 𝟔𝟐 × 𝟎. 𝟑𝟖
𝟎. 𝟎𝟖𝟗 𝟓𝟐𝟖
G
G
G
𝟎. 𝟑𝟖 × 𝟎. 𝟑𝟖 × 𝟎. 𝟑𝟖
𝟎. 𝟎𝟓𝟒 𝟖𝟕𝟐
Note: G stands for “believing in ghosts,” and NG stands for “not believing in ghosts.”
Number of Adults Who Believe in Ghosts
Probability
𝟎
𝟏
𝟐
𝟑
𝟎. 𝟐𝟑𝟖 𝟑𝟐𝟖
𝟎. 𝟒𝟑𝟖 𝟐𝟏𝟔
𝟎. 𝟐𝟔𝟖 𝟓𝟖𝟒
𝟎. 𝟎𝟓𝟒 𝟖𝟕𝟐
Scaffolding:
11. Calculate the probability that at least one adult, but at most two adults, in the sample
believes in ghosts. Interpret this probability in context.
The probability that at least one adult, but at most two adults, believes in ghosts is
𝟎. 𝟒𝟑𝟖 𝟐𝟏𝟔 + 𝟎. 𝟐𝟔𝟖 𝟓𝟖𝟒 = 𝟎. 𝟕𝟎𝟔𝟖.
If three adults were randomly selected and the number of them believing in ghosts was
recorded many times, the proportion that at least one, but at most two, adults believe in
ghosts would be 𝟎. 𝟕𝟎𝟔𝟖.
12. Out of the three randomly selected adults, how many would you expect to believe in
ghosts? Interpret this expected value in context.
MP.2
Out of three randomly selected adults, the expected number who believe in ghosts is as
follows:
𝟎 ∙ 𝟎. 𝟐𝟑𝟖 𝟑𝟐𝟖 + 𝟏 ∙ 𝟎. 𝟒𝟑𝟖 𝟐𝟏𝟔 + 𝟐 ∙ 𝟎. 𝟐𝟔𝟖 𝟓𝟖𝟒 + 𝟑 ∙ 𝟎. 𝟎𝟓𝟒 𝟖𝟕𝟐 = 𝟏. 𝟏𝟒
The long-run average number of adults in a sample of three who believe in ghosts is 𝟏. 𝟏𝟒
adults.
Lesson 10:
Determining Discrete Probability Distributions
This work is derived from Eureka Math ™ and licensed by Great Minds. ©2015 Great Minds. eureka-math.org
This file derived from ALG II-M5-TE-1.3.0-10.2015
 For English language
learners who may struggle
with interpreting
probabilities in context,
consider providing a
sentence frame on the
front board of the
classroom.
 For example:
The long-run average
number of adults in a
sample of _____who
believe in ghosts is _____.
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NYS COMMON CORE MATHEMATICS CURRICULUM
Lesson 10
M5
PRECALCULUS AND ADVANCED TOPICS
Closing (3 minutes)

In the previous lesson, we discussed the purposes of the discrete probability distribution. They were
a.
To make decisions and
b.
To make predictions.
With a partner, discuss whether discrete probabilities are more useful in the short run or in the long run and
why.

Sample response: Discrete probability distributions are more useful in the long run because they
provide the expected distribution after many occurrences of an event rather than just a few. In the
short run, there is too much variability in outcomes for the probability distribution to be of much use.
The law of large numbers tells us that the more times an event occurs, the closer its outcome
distribution will be to the calculated probability distribution.
Remind students that the interpretation of a probability of observing a particular value for a discrete random variable
must include a reference to long-run behavior.
Ask students to summarize the main ideas of the lesson in writing or with a neighbor. Use this as an opportunity to
informally assess comprehension of the lesson. The Lesson Summary below offers some important ideas that should be
included.
Lesson Summary

To derive a discrete probability distribution, you must consider all possible outcomes of the chance
experiment.

The interpretation of probabilities from a probability distribution should mention that it is the long-run
proportion of the time that the corresponding value will be observed.
Exit Ticket (5 minutes)
Lesson 10:
Determining Discrete Probability Distributions
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Lesson 10
NYS COMMON CORE MATHEMATICS CURRICULUM
M5
PRECALCULUS AND ADVANCED TOPICS
Name
Date
Lesson 10: Determining Discrete Probability Distributions
Exit Ticket
23% of the cars a certain automaker manufactures are silver. Below is the probability distribution for the number of
silver cars sold by a car dealer in the next five car sales.
Number of
Silver Cars
Probability
0
1
2
3
4
5
0.27068
0.40426
0.24151
0.07214
0.01077
0.00064
1.
What is the probability of selling at most three silver cars? Interpret this probability in context.
2.
What is the probability of selling between and including one and four silver cars? Interpret this probability in
context.
3.
How many silver cars is the dealer expected to sell, on average, out of five cars? Interpret this expected value in
context.
Lesson 10:
Determining Discrete Probability Distributions
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This file derived from ALG II-M5-TE-1.3.0-10.2015
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Lesson 10
NYS COMMON CORE MATHEMATICS CURRICULUM
M5
PRECALCULUS AND ADVANCED TOPICS
Exit Ticket Sample Solutions
𝟐𝟑% of the cars a certain automaker manufactures are silver. Below is the probability distribution for the number of
silver cars sold by a car dealer in the next five car sales.
Number of
Silver Cars
Probability
1.
𝟎
𝟏
𝟐
𝟑
𝟒
𝟓
𝟎. 𝟐𝟕𝟎𝟔𝟖
𝟎. 𝟒𝟎𝟒𝟐𝟔
𝟎. 𝟐𝟒𝟏𝟓𝟏
𝟎. 𝟎𝟕𝟐𝟏𝟒
𝟎. 𝟎𝟏𝟎𝟕𝟕
𝟎. 𝟎𝟎𝟎𝟔𝟒
What is the probability of selling at most three silver cars? Interpret this probability in context.
The probability that the dealer sells at most three silver cars is as follows:
𝟎. 𝟐𝟕𝟎𝟔𝟖 + 𝟎. 𝟒𝟎𝟒𝟐𝟔 + 𝟎. 𝟐𝟒𝟏𝟓𝟏 + 𝟎. 𝟎𝟕𝟐𝟏𝟒 = 𝟎. 𝟗𝟖𝟖𝟓𝟗.
After many car sales, the long-run proportion of the time that at most three silver cars are sold out of every five is
𝟎. 𝟗𝟖𝟖𝟓𝟗.
2.
What is the probability of selling between and including one and four silver cars? Interpret this probability in
context.
The probability that the dealer sells between one and four silver cars is as follows:
𝟎. 𝟒𝟎𝟒𝟐𝟔 + 𝟎. 𝟐𝟒𝟏𝟓𝟏 + 𝟎. 𝟎𝟕𝟐𝟏𝟒 + 𝟎. 𝟎𝟏𝟎𝟕𝟕 = 𝟎. 𝟕𝟐𝟖𝟔𝟖.
After many car sales, the long-run proportion of the time that between one and four silver cars are sold out of every
five is 𝟎. 𝟕𝟐𝟖𝟔𝟖.
3.
How many silver cars is the dealer expected to sell, on average, out of five cars? Interpret this expected value in
context.
The average number of silver cars the dealer is expected to sell out of five cars is as follows:
𝟎(𝟎. 𝟐𝟕𝟎𝟔𝟖) + 𝟏(𝟎. 𝟒𝟎𝟒𝟐𝟔) + 𝟐(𝟎. 𝟐𝟒𝟏𝟓𝟏) + 𝟑(𝟎. 𝟎𝟕𝟐𝟏𝟒) + 𝟒(𝟎. 𝟎𝟏𝟎𝟕𝟕) + 𝟓(𝟎. 𝟎𝟎𝟎𝟔𝟒) = 𝟏. 𝟏𝟓.
After many car sales, the dealer is expected to sell a long-run average of 𝟏. 𝟏𝟓 silver cars out of every five cars.
Problem Set Sample Solutions
1.
A high school basketball player makes 𝟕𝟎% of the free throws she attempts. Suppose she attempts seven free
throws during a game. The probability distribution for the number of free throws made out of seven attempts is
displayed below.
Number of
Completed
Free Throws
Probability
a.
𝟎
𝟏
𝟐
𝟑
𝟒
𝟓
𝟔
𝟕
𝟎. 𝟎𝟎𝟎𝟐𝟐
𝟎. 𝟎𝟎𝟑𝟓𝟕
𝟎. 𝟎𝟐𝟓𝟎𝟏
𝟎. 𝟎𝟗𝟕𝟐𝟓
𝟎. 𝟐𝟐𝟔𝟖𝟗
𝟎. 𝟑𝟏𝟕𝟔𝟓
𝟎. 𝟐𝟒𝟕𝟎𝟔
𝟎. 𝟎𝟖𝟐𝟑𝟓
What is the probability that she completes at least three free throws? Interpret this probability in context.
The probability that she completes at least three free throws is as follows:
𝟎. 𝟎𝟗𝟕𝟐𝟓 + 𝟎. 𝟐𝟐𝟔𝟖𝟗 + 𝟎. 𝟑𝟏𝟕𝟔𝟓 + 𝟎. 𝟐𝟒𝟕𝟎𝟔 + 𝟎. 𝟎𝟖𝟐𝟑𝟓 = 𝟎. 𝟗𝟕𝟏𝟐.
If this basketball player attempts seven free throws many times, the long-run proportion of the time that she
will complete at least three free throws is 𝟎. 𝟗𝟕𝟐.
Lesson 10:
Determining Discrete Probability Distributions
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Lesson 10
NYS COMMON CORE MATHEMATICS CURRICULUM
M5
PRECALCULUS AND ADVANCED TOPICS
b.
What is the probability that she completes more than two but less than six free throws? Interpret this
probability in context.
The probability that she completes more than two but less than six free throws is as follows:
𝟎. 𝟎𝟗𝟕𝟐𝟓 + 𝟎. 𝟐𝟐𝟔𝟖𝟗 + 𝟎. 𝟑𝟏𝟕𝟔𝟓 = 𝟎. 𝟔𝟒𝟏𝟕𝟗.
If this basketball player attempts seven free throws during a game for many games, the long-run proportion
of the time that she will complete more than two but less than six free throws is 𝟎. 𝟔𝟒𝟏𝟕𝟗.
c.
How many free throws will she complete on average? Interpret this expected value in context.
The average number of free throws that this basketball player will complete out of seven attempts is as
follows:
𝟎(𝟎. 𝟎𝟎𝟎𝟐𝟐) + 𝟏(𝟎. 𝟎𝟎𝟑𝟓𝟕) + 𝟐(𝟎. 𝟎𝟐𝟓𝟎𝟏) + 𝟑(𝟎. 𝟎𝟗𝟕𝟐𝟓) + 𝟒(𝟎. 𝟐𝟐𝟔𝟖𝟗) + 𝟓(𝟎. 𝟑𝟏𝟕𝟔𝟓) +
𝟔(𝟎. 𝟐𝟒𝟕𝟎𝟔) + 𝟕(𝟎. 𝟎𝟖𝟐𝟑𝟓) = 𝟒. 𝟖𝟗𝟗𝟗𝟔 ≈ 𝟒. 𝟗𝟎.
If this basketball player attempts seven free throws many times, the long-run average number of free throws
completed is 𝟒. 𝟗𝟎.
2.
In a certain county, 𝟑𝟎% of the voters are Republicans. Suppose that four voters are randomly selected.
a.
Develop the probability distribution for the random variable number of Republicans out of the four randomly
selected voters.
Voter 1
Voter 2
Voter 3
Voter 4
Calculation
Probability
R
R
R
R
R
R
𝟎. 𝟑 × 𝟎. 𝟑 × 𝟎. 𝟑 × 𝟎. 𝟑
𝟎. 𝟎𝟎𝟖𝟏
R
NR
R
𝟎. 𝟑 × 𝟎. 𝟑 × 𝟎. 𝟑 × 𝟎. 𝟕
𝟎. 𝟎𝟏𝟖𝟗
R
NR
R
𝟎. 𝟑 × 𝟎. 𝟑 × 𝟎. 𝟕 × 𝟎. 𝟑
𝟎. 𝟎𝟏𝟖𝟗
R
NR
R
R
𝟎. 𝟑 × 𝟎. 𝟕 × 𝟎. 𝟑 × 𝟎. 𝟑
𝟎. 𝟎𝟏𝟖𝟗
R
R
NR
NR
𝟎. 𝟑 × 𝟎. 𝟑 × 𝟎. 𝟕 × 𝟎. 𝟕
𝟎. 𝟎𝟒𝟒𝟏
R
NR
R
NR
𝟎. 𝟑 × 𝟎. 𝟕 × 𝟎. 𝟑 × 𝟎. 𝟕
𝟎. 𝟎𝟒𝟒𝟏
R
NR
NR
R
𝟎. 𝟑 × 𝟎. 𝟕 × 𝟎. 𝟕 × 𝟎. 𝟑
𝟎. 𝟎𝟒𝟒𝟏
R
NR
NR
NR
𝟎. 𝟑 × 𝟎. 𝟕 × 𝟎. 𝟕 × 𝟎. 𝟕
𝟎. 𝟏𝟎𝟐𝟗
NR
R
R
R
𝟎. 𝟕 × 𝟎. 𝟑 × 𝟎. 𝟑 × 𝟎. 𝟑
𝟎. 𝟎𝟏𝟖𝟗
NR
R
R
NR
𝟎. 𝟕 × 𝟎. 𝟑 × 𝟎. 𝟑 × 𝟎. 𝟕
𝟎. 𝟎𝟒𝟒𝟏
NR
R
NR
R
𝟎. 𝟕 × 𝟎. 𝟑 × 𝟎. 𝟕 × 𝟎. 𝟑
𝟎. 𝟎𝟒𝟒𝟏
NR
NR
R
R
𝟎. 𝟕 × 𝟎. 𝟕 × 𝟎. 𝟑 × 𝟎. 𝟑
𝟎. 𝟎𝟒𝟒𝟏
NR
R
NR
NR
𝟎. 𝟕 × 𝟎. 𝟑 × 𝟎. 𝟕 × 𝟎. 𝟕
𝟎. 𝟏𝟎𝟐𝟗
NR
NR
R
NR
𝟎. 𝟕 × 𝟎. 𝟕 × 𝟎. 𝟑 × 𝟎. 𝟕
𝟎. 𝟏𝟎𝟐𝟗
NR
NR
NR
R
𝟎. 𝟕 × 𝟎. 𝟕 × 𝟎. 𝟕 × 𝟎. 𝟑
𝟎. 𝟏𝟎𝟐𝟗
NR
NR
NR
NR
𝟎. 𝟕 × 𝟎. 𝟕 × 𝟎. 𝟕 × 𝟎. 𝟕
𝟎. 𝟐𝟒𝟎𝟏
Note: R stands for “Republican,” and NR stands for “not Republican.”
Number of Republicans
Probability
Lesson 10:
𝟎
𝟏
𝟐
𝟑
𝟒
𝟎. 𝟐𝟒𝟎𝟏
𝟎. 𝟒𝟏𝟏𝟔
𝟎. 𝟐𝟔𝟒𝟔
𝟎. 𝟎𝟕𝟓𝟔
𝟎. 𝟎𝟎𝟖𝟏
Determining Discrete Probability Distributions
This work is derived from Eureka Math ™ and licensed by Great Minds. ©2015 Great Minds. eureka-math.org
This file derived from ALG II-M5-TE-1.3.0-10.2015
127
This work is licensed under a
Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License.
Lesson 10
NYS COMMON CORE MATHEMATICS CURRICULUM
M5
PRECALCULUS AND ADVANCED TOPICS
b.
What is the probability that no more than two voters out of the four randomly selected voters will be
Republicans? Interpret this probability in context.
The probability that no more than two out of four randomly selected voters are Republicans is as follows:
𝟎. 𝟐𝟒𝟎𝟏 + 𝟎. 𝟒𝟏𝟏𝟔 + 𝟎. 𝟐𝟔𝟒𝟔 = 𝟎. 𝟗𝟏𝟔𝟑.
If four voters are randomly selected and the number of Republicans is recorded many, many times, the longrun proportion of the time that no more than two voters will be Republicans is 𝟎. 𝟗𝟏𝟔𝟑.
3.
An archery target of diameter 𝟏𝟐𝟐 𝐜𝐦 has a bull’s-eye with diameter 𝟏𝟐. 𝟐 𝐜𝐦.
a.
What is the probability that an arrow hitting the target hits the bull’s-eye?
Area of the bull’s-eye: 𝝅(𝟔. 𝟏 𝐜𝐦)𝟐 = 𝟑𝟕. 𝟐𝟏𝝅 𝐜𝐦𝟐.
Area of the target: 𝝅(𝟔𝟏 𝐜𝐦)𝟐 = 𝟑𝟕𝟐𝟏𝝅 𝐜𝐦𝟐 .
𝟑𝟕. 𝟐𝟏𝝅 𝐜𝐦𝟐
= 𝟎. 𝟎𝟏
𝟑𝟕𝟐𝟏𝝅 𝐜𝐦𝟐
The probability of the arrow landing in the bull’s-eye is 𝟎. 𝟎𝟏.
b.
Develop the probability distribution for the random variable number of bull’s-eyes out of three arrows shot.
Arrow 1
Arrow 2
Arrow 3
Calculation
Probability
NB
NB
NB
𝟎. 𝟗𝟗 × 𝟎. 𝟗𝟗 × 𝟎. 𝟗𝟗
𝟎. 𝟗𝟕𝟎 𝟐𝟗𝟗
NB
NB
B
𝟎. 𝟗𝟗 × 𝟎. 𝟗𝟗 × 𝟎. 𝟎𝟏
𝟎. 𝟎𝟎𝟗 𝟖𝟎𝟏
NB
B
NB
𝟎. 𝟗𝟗 × 𝟎. 𝟎𝟏 × 𝟎. 𝟗𝟗
𝟎. 𝟎𝟎𝟗 𝟖𝟎𝟏
NB
B
B
𝟎. 𝟗𝟗 × 𝟎. 𝟎𝟏 × 𝟎. 𝟎𝟏
𝟎. 𝟎𝟎𝟎 𝟎𝟗𝟗
B
NB
NB
𝟎. 𝟎𝟏 × 𝟎. 𝟗𝟗 × 𝟎. 𝟗𝟗
𝟎. 𝟎𝟎𝟗 𝟖𝟎𝟏
B
B
NB
𝟎. 𝟎𝟏 × 𝟎. 𝟎𝟏 × 𝟎. 𝟗𝟗
𝟎. 𝟎𝟎𝟎 𝟎𝟗𝟗
B
NB
B
𝟎. 𝟎𝟏 × 𝟎. 𝟗𝟗 × 𝟎. 𝟎𝟏
𝟎. 𝟎𝟎𝟎 𝟎𝟗𝟗
B
B
B
𝟎. 𝟎𝟏 × 𝟎. 𝟎𝟏 × 𝟎. 𝟎𝟏
𝟎. 𝟎𝟎𝟎 𝟎𝟎𝟏
Note: B stands for “bull’s-eye,” and NB stands for “not bull’s-eye.”
Number of Bull’s-Eyes
Probability
c.
𝟎
𝟏
𝟐
𝟑
𝟎. 𝟗𝟕𝟎 𝟐𝟗𝟗
𝟎. 𝟎𝟐𝟗 𝟒𝟎𝟑
𝟎. 𝟎𝟎𝟎 𝟐𝟗𝟕
𝟎. 𝟎𝟎𝟎 𝟎𝟎𝟏
What is the probability of an archer getting at least one bull’s-eye? Interpret this probability in context.
𝟎. 𝟎𝟐𝟗 𝟒𝟎𝟑 + 𝟎. 𝟎𝟎𝟎 𝟐𝟗𝟕 + 𝟎. 𝟎𝟎𝟎 𝟎𝟎𝟏 = 𝟎. 𝟎𝟐𝟗 𝟕𝟎𝟏
After shooting many arrows, the long-run proportion of an archer getting at least one bull’s-eye out of three
is 𝟎. 𝟎𝟐𝟗 𝟕𝟎𝟏.
d.
On average, how many bull’s-eyes should an archer expect out of three arrows? Interpret this expected value
in context.
𝟎(𝟎. 𝟗𝟕𝟎 𝟐𝟗𝟗) + 𝟏(𝟎. 𝟎𝟐𝟗 𝟒𝟎𝟑) + 𝟐(𝟎. 𝟎𝟎𝟎 𝟐𝟗𝟕) + 𝟑(𝟎. 𝟎𝟎𝟎 𝟎𝟎𝟏) = 𝟎. 𝟎𝟑
After shooting many arrows, the long-run average should be close to 𝟎. 𝟎𝟑 bull’s-eyes for every three arrows
shot.
Lesson 10:
Determining Discrete Probability Distributions
This work is derived from Eureka Math ™ and licensed by Great Minds. ©2015 Great Minds. eureka-math.org
This file derived from ALG II-M5-TE-1.3.0-10.2015
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This work is licensed under a
Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License.
Lesson 10
NYS COMMON CORE MATHEMATICS CURRICULUM
M5
PRECALCULUS AND ADVANCED TOPICS
4.
The probability that two people have the same birthday in a room of 𝟐𝟎 people is about 𝟒𝟏. 𝟏%. It turns out that
your math, science, and English classes all have 𝟐𝟎 people in them.
a.
Develop the probability distribution for the random variable number of pairs of people who share birthdays
out of three classes.
Math
Science
English
Calculation
Probability
NP
NP
NP
𝟎. 𝟓𝟖𝟗 × 𝟎. 𝟓𝟖𝟗 × 𝟎. 𝟓𝟖𝟗
𝟎. 𝟐𝟎𝟒 𝟑𝟑𝟔
NP
NP
P
𝟎. 𝟓𝟖𝟗 × 𝟎. 𝟓𝟖𝟗 × 𝟎. 𝟒𝟏𝟏
𝟎. 𝟏𝟒𝟐 𝟓𝟖𝟓
NP
P
NP
𝟎. 𝟓𝟖𝟗 × 𝟎. 𝟒𝟏𝟏 × 𝟎. 𝟓𝟖𝟗
𝟎. 𝟏𝟒𝟐 𝟓𝟖𝟓
NP
P
P
𝟎. 𝟓𝟖𝟗 × 𝟎. 𝟒𝟏𝟏 × 𝟎. 𝟒𝟏𝟏
𝟎. 𝟎𝟗𝟗 𝟒𝟗𝟒
P
NP
NP
𝟎. 𝟒𝟏𝟏 × 𝟎. 𝟓𝟖𝟗 × 𝟎. 𝟓𝟖𝟗
𝟎. 𝟏𝟒𝟐 𝟓𝟖𝟓
P
P
NP
𝟎. 𝟒𝟏𝟏 × 𝟎. 𝟒𝟏𝟏 × 𝟎. 𝟓𝟖𝟗
𝟎. 𝟎𝟗𝟗 𝟒𝟗𝟒
P
NP
P
𝟎. 𝟒𝟏𝟏 × 𝟎. 𝟓𝟖𝟗 × 𝟎. 𝟒𝟏𝟏
𝟎. 𝟎𝟗𝟗 𝟒𝟗𝟒
P
P
P
𝟎. 𝟒𝟏𝟏 × 𝟎. 𝟒𝟏𝟏 × 𝟎. 𝟒𝟏𝟏
𝟎. 𝟎𝟔𝟗 𝟒𝟐𝟕
Note: P stands for “pair,” and NP stands for “no pair.”
Number of Pairs of
Birthday Sharers
Probability
b.
𝟎
𝟏
𝟐
𝟑
𝟎. 𝟐𝟎𝟒 𝟑𝟑𝟔
𝟎. 𝟒𝟐𝟕 𝟕𝟓𝟓
𝟎. 𝟐𝟗𝟖 𝟒𝟖𝟐
𝟎. 𝟎𝟔𝟗 𝟒𝟐𝟕
What is the probability that one or more pairs of people share a birthday in your three classes? Interpret the
probability in context.
𝟎. 𝟒𝟐𝟕 𝟕𝟓𝟓 + 𝟎. 𝟐𝟗𝟖 𝟒𝟖𝟐 + 𝟎. 𝟎𝟔𝟗 𝟒𝟐𝟕 = 𝟎. 𝟕𝟗𝟓 𝟔𝟔𝟒
If you went to many classes or other events containing 𝟐𝟎 people, the long-run proportion of groups in which
at least one pair of people share the same birthday is 𝟎. 𝟕𝟗𝟓 𝟔𝟔𝟒.
5.
You go to the warehouse of the computer company you work for because you need to send eight motherboards to a
customer. You realize that someone has accidentally reshelved a pile of motherboards you had set aside as
defective. Thirteen motherboards were set aside, and 𝟏𝟕𝟐 are known to be good. You are in a hurry, so you pick
eight at random. The probability distribution for the number of defective motherboards is below.
Number of
Defective
Motherboards
Probability
a.
𝟎
𝟏
𝟐
𝟑
𝟒
𝟓
𝟔
𝟕
𝟖
𝟎. 𝟓𝟓𝟗𝟔
𝟎. 𝟑𝟑𝟕𝟎
𝟎. 𝟎𝟖𝟖𝟖
𝟎. 𝟎𝟏𝟑𝟒
𝟎. 𝟎𝟎𝟏𝟑
𝟎. 𝟑𝟏𝟕𝟕
𝟕. 𝟔 × 𝟏𝟎−𝟓
𝟔. 𝟏 × 𝟏𝟎−𝟖
𝟓. 𝟕𝟔 × 𝟏𝟎−𝟏𝟎
If more than one motherboard is defective, your company may lose the customer’s business. What is the
probability of that happening?
𝟏 − (𝟎. 𝟓𝟓𝟗𝟔 + 𝟎. 𝟑𝟑𝟕𝟎) = 𝟎. 𝟏𝟎𝟑𝟒
b.
You are in a hurry and get nervous, so you pick eight motherboards and then second-guess yourself and put
them back on the shelf. You then pick eight more. You do this a few times and then decide it is time to make
a decision and send eight motherboards to the customer. On average, how many defective motherboards are
you choosing each time? Is it worth the risk of blindly picking motherboards?
𝟎(𝟎. 𝟓𝟓𝟗𝟔) + 𝟏(𝟎. 𝟑𝟑𝟕𝟎) + 𝟐(𝟎. 𝟎𝟖𝟖𝟖) + 𝟑(𝟎. 𝟎𝟏𝟑𝟒) + 𝟒(𝟎. 𝟎𝟎𝟏𝟑) + 𝟓(𝟎. 𝟑𝟏𝟕𝟕) + 𝟔(𝟕. 𝟔 × 𝟏𝟎−𝟓 ) +
𝟕(𝟔. 𝟏 × 𝟏𝟎−𝟖 ) + 𝟖(𝟓. 𝟕𝟔 × 𝟏𝟎−𝟏𝟎 ) = 𝟐. 𝟏𝟓
On average, 𝟐. 𝟏𝟓 motherboards are defective, which would suggest that you run the risk of losing the
customer’s business.
Lesson 10:
Determining Discrete Probability Distributions
This work is derived from Eureka Math ™ and licensed by Great Minds. ©2015 Great Minds. eureka-math.org
This file derived from ALG II-M5-TE-1.3.0-10.2015
129
This work is licensed under a
Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License.