# STATS Modeling The World

## Educators

CC

### Problem 1

For each of the following scenarios, decide if the outcome is random.
a) Flip a coin to decide who takes out the trash. Is who takes out the trash random?
b) A friend asks you to quickly name a professional sports team. Is the sports team named random?
c) Names are selected out of a hat to decide roommates in a dormitory. Is your roommate for the year random?

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### Problem 2

For each of the following scenarios, decide if the outcome is random.
a) You enter a contest in which the winning ticket is selected from a large drum of entries. Was the winner of the contest random?
b) When playing a board game, the number of spaces you move is decided by rolling a six-sided die. Is the number of spaces you move random?

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### Problem 3

Many states run lotteries, giving away millions of dollars if you match a certain set of winning numbers. How are those numbers determined? Do you think this method guarantees randomness? Explain.

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### Problem 4

Many kinds of games people play rely on randomness. Cite three different methods commonly used in the attempt to achieve this randomness, and discuss the effectiveness of each.

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### Problem 5

The American College of Obstetricians and Gynecologists says that out of every 100 babies born in the United States, 3 have some kind of major birth defect. How would you assign random numbers to conduct a simulation based on this statistic?

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### Problem 6

By some estimates, about 10% of all males have some color perception defect, most commonly red–green colorblindness. How would you assign random numbers to conduct a simulation based on this statistic?

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### Problem 7

An elementary school teacher with 25 students plans to have each of them make a poster about two different states. The teacher first numbers the states (in alphabetical order, from 01-Alabama to 50-Wyoming), then uses a random number table to decide which states each kid gets. Here are the random digits:
$$45921017102289237076$$
a) Which two state numbers does the first student get?
b) Which two state numbers go to the second student?

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### Problem 31

Many couples want to have both a boy and a girl. If they decide to continue to have children until they have one child of each sex, what would the average family size be? Assume that boys and girls are equally likely.

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### Problem 32

Suppose a couple will continue having children until they have at least two children of each sex (two boys and two girls). How many children might they expect to have?

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### Problem 33

You are playing a children’s game in which the number of spaces you get to move is determined by the rolling of a die. You must land exactly on the final space in order to win. If you are 10 spaces away, how many turns might it take you to win?

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### Problem 34

You are three spaces from a win in Parcheesi. On each turn, you will roll two dice. To win, you must roll a total of 3 or roll a 3 on one of the dice. How many turns might you expect this to take?

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### Problem 35

A basketball player with a 65% shooting percentage has just made 6 shots in a row. The announcer
says this player “is hot tonight! She’s in the zone!” Assume the player takes about 20 shots per game. Is it unusual for her to make 6 or more shots in a row during a game?

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### Problem 36

The World Series ends when a team wins 4 games. Suppose that sports analysts consider one team a bit stronger, with a 55% chance to win any individual game. Estimate the likelihood that the underdog wins the series.

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### Problem 37

Four couples at a dinner party play a board game after the meal. They decide to play as teams of two and to select the teams randomly. All eight people write their names on slips of paper. The slips are thoroughly mixed, then drawn two at a time. How likely is it that every person will be teamed with someone other than the person he or she came to the party with?

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### Problem 38

Suppose the couples in Exercise 37 choose the teams by having one member of each couple write their names on the cards and the other people each pick a card at random. How likely is it that every person will be teamed with someone other than the person he or she came with?

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### Problem 39

A company with a large sales staff announces openings for three positions as regional managers. Twenty-two of the current salespersons apply, 12 men and 10 women. After the interviews, when the company announces the newly appointed managers, all three positions go to women. The men complain of job discrimination. Do they have a case? Simulate a random selection of three people from the applicant pool, and make a decision about the likelihood that a fair process would result in hiring all women.

CC
Charles C.