## Engage NY Eureka Math Precalculus Module 5 Lesson 5 Answer Key

### Eureka Math Precalculus Module 5 Lesson 5 Exercise Answer Key

Exercise 1.

1. Sort the features of each apartment into three categories:

Answer:

a. Describe how the features listed in each category are similar.

Answer:

The number of bedrooms and floor numbers are integers.

The size of the apartment and distance to the elevator are not all integers, and none of the values are the same.

The color of the walls and floor type are verbal descriptions, not numbers.

b. A random variable associates a number with each outcome of a chance experiment. Which of the features are random variables? Explain.

Exercises 2–3

Exercise 2.

For each of the six variables listed in Exercise 1, give a specific example of a possible value the variable might have taken on, and identify the variable as discrete or continuous.

Answer:

Responses will vary.

The number of bedrooms is a discrete random variable; a possible value of this variable is 3. The distance to the elevator could be 100 ft., and it is a continuous variable because it could be a little more or less than 100 ft. depending on where your starting point is within the apartment. The discrete random variables are number of bedrooms, floor number, color, and floor type.

Exercise 3.

Suppose you were collecting data about dogs. Give at least two examples of discrete and two examples of continuous data you might collect.

Answer:

Responses will vary.

Continuous data: length of tail, length of ears, height, weight

Discrete data: number of legs, typical number of puppies in the litter, whether ears point up or down or break in the middle and flop

Exercises 4–8: Music Genres

People like different genres of music: country, rock, hip – hop, jazz, and so on. Suppose you were to give a survey to people asking them how many different music genres they like.

Exercise 4.

What do you think the possible responses might be?

Answer:

Possible answer: 0, 1, 2, etc.

Exercise 5.

The table below shows 11,565 responses to the survey question: How many music genres do you like listening to?

Table 1: Number of Music Genres Survey Responders Like Listening To

Find the relative frequency for each possible response (each possible value for number of music genres), rounded to the nearest hundredth. (The relative frequency is the proportion of the observations that take on a particular value. For example, the relative frequency for 0 is \(\frac{568}{11565}\).)

Answer:

Note: Due to rounding, values may not always add up to exactly 1.

Exercise 6.

Consider the chance experiment of selecting a person at random from the people who responded to this survey. The table you generated in Exercise 5 displays the probability distribution for the random variable number of music genres liked. Your table shows the different possible values of this variable and the probability of observing each value.

a. Is the random variable discrete or continuous?

Answer:

The random variable is discrete because the possible values are 0, 1, …, 8, and these are isolated points along the number line.

b. What is the probability that a randomly selected person who responded to the survey said that she likes 3 different music genres?

Answer:

0.06

c. Which of the possible values of this variable has the greatest probability of being observed?

Answer:

The greatest probability is 8 genres, which has a probability of 0.25.

d. What is the probability that a randomly selected person who responded to the survey said that he liked 1 or fewer different genres?

Answer:

0.22

e. What is the sum of the probabilities of all of the possible outcomes? Explain why your answer is reasonable for the situation.

Answer:

1.00 or close to 1.00. The probabilities of all the possible values should add to up to 1 because they represent everything that might possibly occur. However, due to rounding, values may not always add up to exactly 1.

Exercise 7.

The survey data for people age 60 or older are displayed in the graphs below.

a. What is the difference between the two graphs?

Answer:

The graph on the left shows the total number of people (the frequency) for each possible value of the random variable number of music genres liked. The graph on the right shows the relative frequency for each possible value.

b. What is the probability that a randomly selected person from this group of people age 60 or older chose 4 music genres?

Answer:

0.08

c. Which of the possible values of this variable has the greatest probability of occurring?

Answer:

One genre, with a probability of 0.30

d. What is the probability that a randomly selected person from this group of people age 60 or older chose 5 different genres?

Answer:

0

e. Make a conjecture about the sum of the relative frequencies. Then, check your conjecture using the values in the table.

Answer:

Responses will vary.

The sum should be 1 because that would be the total of the probabilities of all of the outcomes:

0.07 + 0.30 + 0.17 + 0.02 + 0.09 + 0.27 + 0.07 = 0.99, which is not quite 1, but there is probably some rounding error. Note that students might not get the exact values when reading off the graph, but their answers should be close.

Exercise 8.

Below are graphs of the probability distribution based on responses to the original survey and based on responses from those age 60 and older.

Identify which of the statements are true and which are false. Give a reason for each claim.

a. The probability that a randomly selected person chooses 0 genres is greater for those age 60 and older than for the group that responded to the original survey.

Answer:

True: Overall, the probability is about 0.05, and for those 60 and older, it is about 0.07.

b. The probability that a randomly selected person chooses fewer than 3 genres is smaller for those age 60 and older than for the group that responded to the original survey.

Answer:

False: Overall, the probability is 0.35, and for those 60 and older, it is 0.54.

c. The sum of the probabilities for all of the possible outcomes is larger for those age 60 and older than for the group that responded to the original survey.

Answer:

False: In both cases, the sum of the probabilities is 1.

Exercises 9–11: Family Sizes

The table below displays the distribution of the number of people living in a household according to a recent U.S. Census. This table can be thought of as the probability distribution for the random variable that consists of recording the number of people living in a randomly selected U.S. household. Notice that the table specifies the possible values of the variable, and the relative frequencies can be interpreted as the probability of each of the possible values.

Table 2: Relative Frequency of the Number of People Living in a Household

Exercise 9.

What is the random variable, and is it continuous or discrete? What values can it take on?

Answer:

The random variable is the number of people in a household, and it is discrete. The possible values are 1, 2, 3, 4, 5, 6, 7, or more.

Exercise 10.

Use the table to answer each of the following:

a. What is the probability that a randomly selected household would have 5 or more people living there?

Answer:

0.07 + 0.02 + 0.02 = 0.11

b. What is the probability that 1 or more people live in a household? How does the table support your answer?

Answer:

Common sense says that 100% of the households should have 1 or more people living in them. If you add up the relative frequencies for the different numbers of people per household, you get 1.00.

c. What is the probability that a randomly selected household would have fewer than 6 people living there? Find your answer in two different ways.

Answer:

By adding the probabilities for 1, 2, 3, 4, and 5 people in a household, the answer would be 0.96.

By adding the probabilities for 6 and 7 or more people living in a household and then subtracting the sum from 1, the answer would be 1 – 0.04 = 0.96.

Exercise 11.

The probability distributions for the number of people per household in 1790, 1890, and 1990 are below.

Source: U.S. Census Bureau (www.census.gov)

a. Describe the change in the probability distribution of the number of people living in a randomly selected household over the years.

Answer:

Responses will vary.

In 1790 and 1890, the largest percentage of people were living in households of 7 or more people. In 1990, most people lived in houses with 1 or 2 people.

b. What are some factors that might explain the shift?

Answer:

Responses will vary.

The shift might be because more people lived in urban areas instead of rural areas in the 1990s; more extended families with parents and grandparents lived in the same household in the 1790s and 1890s; more children lived in the same household per family in the earlier years.

### Eureka Math Precalculus Module 5 Lesson 5 Problem Set Answer Key

Question 1.

Each person in a large group of children with cell phones was asked, “How old were you when you first received a cell phone?”

The responses are summarized in the table below.

a. Make a graph of the probability distribution.

Answer:

b. The bar centered at 12 in your graph represents the probability that a randomly selected person in this group first received a cell phone at age 12. What is the area of the bar representing age 12? How does this compare to the probability corresponding to 12 in the table?

Answer:

The base of the rectangle is 1, and the height is 0.23, so the area should be 0.23. This is the same as the probability for 12 in the table.

c. What do you think the sum of the areas of all of the bars will be? Explain your reasoning.

Answer:

The sum of all the areas should be 1 because the sum of all probabilities in the probability distribution of a discrete random variable is always 1 or very close to 1 due to rounding.

d. What is the probability that a randomly selected person from this group first received a cell phone at age 12 or 13?

Answer:

0.46

e. Is the probability that a randomly selected person from this group first received a cell phone at an age older than 15 greater than or less than the probability that a randomly selected person from this group first received a cell phone at an age younger than 12?

Answer:

P(older than 15) = 0.09; p(< 12) = 0.20; the probability for over 15 is less than the probability for under 12.

Question 2.

The following table represents a discrete probability distribution for a random variable. Fill in the missing values so that the results make sense; then, answer the questions.

Answer:

Responses will vary.

The two missing values can be any two positive numbers whose sum adds to 0.33. For example, the probability for 5 could be 0.03, and the probability for 15 could be 0.3.

a. What is the probability that this random variable takes on a value of 4 or 5?

Answer:

Responses will vary.

Possible answer: 0.08 + 0.03 = 0.11

b. What is the probability that the value of the random variable is not 15?

Answer:

Responses will vary.

Possible answer: 1 – 0.3 = 0.7

c. Which possible value is least likely?

Answer:

Responses will vary.

Possible answer: 5 would be the least likely as it has the smallest probability.

Question 3.

Identify the following as true or false. For those that are false, explain why they are false.

a. The probability of any possible value in a discrete random probability distribution is always greater than or equal to 0 and less than or equal to 1.

Answer:

True

b. The sum of the probabilities in a discrete random probability distribution varies from distribution to distribution.

Answer:

False; the sum of the probabilities is always equal to 1 or very close to 1 due to rounding.

c. The total number of times someone has moved is a discrete random variable.

Answer:

True

Question 4.

Suppose you plan to collect data on your classmates. Identify three discrete random variables and three continuous random variables you might observe.

Answer:

Responses will vary. Possible responses are shown below.

Discrete: how many siblings; how many courses they are taking; how many pets they have in their home; how many cars are in their family; how many movies they saw last month

Continuous: height; handspan; time it takes to get to school; time per week playing video games

Question 5.

Which of the following are not possible for the probability distribution of a discrete random variable? For each one you identify, explain why it is not a legitimate probability distribution.

Answer:

The first distribution cannot be a probability distribution because the given probabilities add to more than 1. The second distribution cannot be a probability distribution because there is a negative probability given, and probabilities cannot be negative.

Question 6.

Suppose that a fair coin is tossed 2 times, and the result of each toss (H or T) is recorded.

a. What is the sample space for this chance experiment?

Answer:

{HH, HT, TH, TT}

b. For this chance experiment, give the probability distribution for the random variable of the total number of heads observed.

Answer:

Question 7.

Suppose that a fair coin is tossed 3 times.

a. How are the possible values of the random variable of the total number of heads observed different from the possible values in the probability distribution of Problem 6(b)?

Answer:

Possible values are now 0, 1, 2, and 3.

b. Is the probability of observing a total of 2 heads greater when the coin is tossed 2 times or when the coin is tossed 3 times? Justify your answer.

Answer:

The probability of 2 heads is greater when the coin is tossed 3 times. The probability distribution of the number of heads for 3 tosses is

The probability for the possible value of 2 is 0.375 for 3 tosses and only 0.25 for 2 tosses.

### Eureka Math Precalculus Module 5 Lesson 5 Exit Ticket Answer Key

Question 1.

Create a table that illustrates the probability distribution of a discrete random variable with four outcomes.

Answer:

Answers will vary.

One possible answer:

Check to make sure that all probabilities are between 0 and 1 and that the probabilities add to 1.

Question 2.

Which of the following variables are discrete, and which are continuous? Explain your answers.

Number of items purchased by a customer at a grocery store

Time required to solve a puzzle

Length of a piece of lumber

Number out of 10 customers who pay with a credit card

Answer:

Discrete: number of items purchased by a customer at a grocery store; number out of 10 customers who pay with a credit card; they can each be described by an integer value.

Continuous: time required to solve a puzzle; length of a piece of lumber; the values need to be measured and involve intervals along a number line.