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AP Review – Random Variables & Binomial Distributions

1) Which of the following are true statements?

I. The histogram of a binomial distribution with p = .5 is always symmetric no matter what the value of n is.

II. The histogram of a binomial distribution with p = .2 is skewed to the left.

III. The histogram of a binomial distribution with p = .9 looks more and more symmetric, the larger the value of n.

A) I and II B) I and III C) II and III

D) I, II, and III E) None of the above

2) A television game show has three payoffs with the following probabilities:

Payoff ($) 0 1000 10,000

Probability .6 .3 .1

What are the mean and standard deviation for the payoff variable?

A) (x = 1300, (x = 2934

B) (x = 1300, (x = 8802

C) (x = 3667, (x = 4497

D) (x = 3667, (x = 5508

E) None of the above gives a set of correct answers.

3) At a warehouse sale, 100 customers are invited to choose one of 100 identical boxes. Five boxes contain $700 color television sets, 25 boxes contain $540 camcorders, and the remaining boxes contain $260 cameras. What should a customer be willing to pay to participate?

A) $260 B) $352

C) $500 D) $540

E) $699

4) If the probability of a basketball player scoring on any shot is .75, what is the probability that she scores on at most 5 of her next 6 shots?

A) .3560 B) .8220 C) .1780

D) .4661 E) .5339

5) The average annual incomes of high school and college graduates in a Midwestern town are $21,000 and $35,000, respectively. If a company hires only personnel with at least a high school diploma and 20% of its employees have been through college, what is the mean income of the company employees?

A) $23,800 B) $27,110 C) $28,000

D) $32,200 E) $56,000

6) An insurance company charges $800 annually for car insurance. The policy specifies that the company will pay $1000 for a minor accident and $5000 for a major accident. If the probability of a motorist having a minor accident during the year is .2, and of having a major accident, .05, how much can the insurance company expect to make on the policy?

A) $200 B) $250 C) $300

D) $350 E) $450

7) Find the probability that a family of five children will have exactly three boys.

A) .3125 B) .6875 C) .8125

D) .1875 E) .1563

8) Which of the following are true statements?

I. By the law of large numbers, the mean of a random variable will get closer and closer to a specific value.

II. The standard deviation of a random variable is never negative.

III. The standard deviation of a random variable is 0 only if the random variable takes a single value.

A) I and II B) I and III C) II and III

D) I, II, and III E) None of the above

9) Mathematically speaking, casinos and life insurance companies make a profit because of

A) their understanding of sampling error and sources of bias

B) their use of well-designed, well-conducted surveys and experiments

C) their use of simulation of probability distributions

D) the central limit theorem

E) the law of large numbers

10) There are 8,253 men and 10,327 women at a state university. If 43% of men and 27% of women are business majors, what is the expected number of business majors in a random sample of 200 students?

a) 31.7 b) 34.1 c) 63.4

d) 68.2 e) 70.0

11) You have a choice of three investments, the first of which gives you a 10% chance of making $1 million, otherwise you lose $25,000; the second of which gives you a 50% chance of making $500,000, otherwise you lose $345,000; and the third of which gives you an 80% chance of making $50,000, otherwise you make only $1000. Assuming you will go bankrupt it you don’t show a profit, which option should you choose for the best chance of avoiding bankruptcy?

A) First choice B) Second choice

C) Third choice D) Either the first or second choice

E) All the choices give an equal chance of avoiding bankruptcy

12) Can the function [pic] for x = 1, 2, and 3, be the probability distribution for some random variable?

A) Yes

B) No, because probabilities cannot be negative

C) No, because probabilities cannot be greater than 1

D) No, because the probabilities do not sum to 1

E) Not enough information is given to answer the question

13) Which of the following is a discrete random variable?

A) The number of times a student guesses answers on a test

B) The amount of gasoline purchased by a customer

C) The amount of mercury found in fish caught in the Gulf of Mexico

D) The height of water-oak trees

E) The time elapsed until the first field goal at home football games

14) Suppose you are one of 7.5 million people who send in their name for a drawing with 1 top prize of $1 million, 5 second-place prizes of $10,000, and 20 third-place prizes of $100. Is it worth the $0.37 postage it cost you to send in your name?

A) Yes, because [pic], which is less than 7,500,000

B) No, because your expected winnings are only $0.14

C) Yes, because [pic]

D) No, because 1,052,000 < 7,500,000

E) Yes because [pic]

15) Suppose X and Y are random variables with E(X) = 500, Var(X) = 50, E(Y) = 400, and Var(Y) = 30. Given that X and Y are independent, what are the expected value and variance of the random variable X – Y?

A) E(X – Y) = 100, Var(X – Y) = 20

B) E(X – Y) = 100, Var(X – Y) = 80

C) E(X – Y) = 900, Var(X – Y) = 20

D) E(X – Y) = 900, Var(X – Y) = 80

E) There is insufficient information to answer this question.

16) A medicine is known to produce side effects in 1 in 5 patients taking it. Suppose a doctor prescribes the medicine to 4 unrelated patients. What is the probability that none of the patients will develop side effects?

A) 0.8 B) 0.4096 C) 0.25

D) 0.2 E) 0.0016

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