## Engage NY Eureka Math Precalculus Module 5 Lesson 16 Answer Key

### Eureka Math Precalculus Module 5 Lesson 16 Exploratory Challenge Answer Key

Exploratory Challenge 2

Work with your group to explore each of the decision – making methods. Your teacher will assign one or more methods to each group and specify the number of times each decision should be simulated. Record the outcomes in the table.

Which method do you think should be used to make a fair decision in this case? Explain.

Answer:

Answers will vary.

### Eureka Math Precalculus Module 5 Lesson 16 Problem Set Answer Key

Question 1.

You and your sister each want to sit in the front seat of your mom’s car. For each of the following, decide if the decision would be fair or unfair, and explain your answer.

a. You flip a two – sided coin.

Answer:

Answers may vary. Some students may say fair as it may be a two – sided coin with a 50/50 probability. Others may say unfair as the person flipping the coin could use a coin with two heads, which would favor one person over the other.

b. Both you and your sister try to pick a number closest to one randomly generated on a smartphone.

Answer:

Fair. Neither person is favored over the other.

c. You let your mom decide.

Answer:

Unfair. For example, you may not have cleaned your room that day, and your mom may favor your sister instead

Question 2.

Janice, Walter, and Brooke are siblings. Their parents need them to divide the chores around the house. The one task no one wants to volunteer for is cleaning the bathroom. Janice sees a deck of 52 playing cards sitting on the table and convinces her brother and sister to use the cards to decide who will clean the bathroom.

- Janice thinks they should draw one card. If a heart is drawn, Janice cleans the bathroom. If a spade is drawn, then Walter cleans. If a diamond is drawn, then Brooke cleans. All of the club cards will be removed from the deck before they begin drawing a card.
- Walter wants to draw two cards at a time. If both cards are red, then Janice cleans. If both cards are black, then Walter cleans. If one card is red and one card is black, then Brooke cleans the bathroom.
- Brooke thinks they should draw cards until they get a heart. If the first card drawn is a heart, then Janice cleans the bathroom. If the second card drawn is a heart, then Walter cleans the bathroom. If it takes three or more times to draw a heart, then Brooke cleans the bathroom.

Whose method is fair? Explain using probabilities.

Answer:

Janice’s method is fair. They each have an equal probability of being chosen to clean the bathroom: \(\frac{13}{29}\) or \(\frac{1}{3}\).

Walter’s method is not fair. The probability of drawing two red cards is \(\frac{26}{52}\)∙\(\frac{25}{51}\) ≈ 0.245. The probability of drawing two black cards is the same, \(\frac{26}{52}\)∙\(\frac{25}{51}\) ≈ 0.245. The probability of drawing a red card and a black card is 1 – 0.245 – 0.245 ≈ 0.51. So, Brooke is more likely to clean this bathroom using this method.

Brooke’s method is not fair. The probability of the first card drawn being a heart is \(\frac{13}{52}\) = 0.25. The probability of the second card drawn being a heart is \(\frac{39}{52}\)∙\(\frac{13}{51}\) ≈ 0.191. The probability that it takes three or more cards to draw a heart is 1 – 0.25 – 0.191 ≈ 0.559. So, Brooke is more likely to clean this bathroom using this method.

Question 3.

A large software company is moving into new headquarters. Although the workspace is larger, there is not enough space for each of the 239 employees to have his own office. It turns out that two of the employees will need to share an office. Explain how to use a random number generator to make a fair decision as to which employees will share an office.

Answer:

Answers will vary. Assign each of the employees a number, 1 to 239. Then, generate two random numbers between 1 and 239. The employees corresponding to these numbers will share an office.

Question 4.

Leslie and three of her friends each want to eat dinner at different restaurants. Describe a fair way to decide to which restaurant the four friends should go to eat dinner.

Answer:

Answers will vary. Each of the four friends can write the name of the restaurant they prefer on a slip of paper (all slips the same) and draw a slip out of a hat.

### Eureka Math Precalculus Module 5 Lesson 16 Exit Ticket Answer Key

Question 1.

Both Carmen and her brother Michael want to borrow their father’s car on a Friday night. To determine who gets to use the car, Carmen wants her father to roll a pair of fair dice. If the sum of the two dice is 2, 3, 11, or 12, Carmen gets to use the car. If the sum of the two dice is 4 or 10, then Michael can use the car. If the sum is any other number, then the dice will be rolled again. Michael thinks that this is not a fair way to decide. Is he correct? Explain.

Answer:

No. Michael is wrong. The probability of rolling a sum of a 2, 3, 11, or 12 is \(\frac{1}{21}\) + \(\frac{1}{21}\) + \(\frac{1}{21}\) + \(\frac{1}{21}\) = \(\frac{4}{21}\).

The probability of rolling a sum of a 4 or 10 is \(\frac{2}{21}\) + \(\frac{2}{21}\) = \(\frac{4}{21}\). Because both Carmen and Michael are equally likely to get the car, this would yield a fair decision.

Question 2.

Due to a technology glitch, an airline has overbooked the number of passengers in economy class on a flight from New York City to Los Angeles. Currently, there are 150 passengers who have economy class tickets, but there are only 141 seats on the plane. There are two seats available in first class and one seat available in business class.

a. Explain how the ticket agent could use a random number generator to make a fair decision in moving some passengers to either the first – or business – class sections of the plane and to rebook the extra passengers to a later flight.

Answer:

The ticket agent can assign each of the 150 passengers a number, 1 through 150. Since there are nine extra passengers, the ticket agent can generate nine random numbers between 1 and 150. The first two numbers will be moved to first class, the third number will be moved to business class, and the remaining six numbers will need to be moved to another flight. (If any random numbers repeat, then generate additional numbers as needed.)

b. Is there any other way for the ticket agent to make a fair decision? Explain.

Answer:

The ticket agent could draw numbered balls from a bag. For example, use balls numbered 1 through 150, and choose one ball at a time.