# Mastery 4

1.

India is the second most populous country in the world, with a population of over 1 billion people. Although the government has offered various incentives for population control, some argue that the birth rate, especially in rural India, is still too high to be sustainable. A demographer assumes the following probability distribution of the household size in India.

Household Size Probability

1 0.06

2 0.13

3 0.17

4 0.22

5 0.18

6 0.14

7 0.08

8 0.02

a.

What is the probability that there are less than 5 members in a typical household in India? (Round your answer to 2 decimal places.)

Probability

b.

What is the probability that there are 5 or more members in a typical household in India? (Round your answer to 2 decimal places.)

Probability

c.

What is the probability that the number of members in a typical household in India is greater than 5 and less than 8 members? (Round your answer to 2 decimal places.)

Probability

rev: 02_26_2014_QC_45094

2.

Market observers are quite uncertain whether the stock market has bottomed out from the economic meltdown that began in 2008. In an interview on March 8, 2009, CNBC interviewed two prominent economists who offered differing views on whether the U.S. economy was getting stronger or weaker. An investor not wanting to miss out on possible investment opportunities considers investing $12,000 in the stock market. He believes that the probability is 0.23 that the market will improve, 0.41 that it will stay the same, and 0.36 that it will deteriorate. Further, if the economy improves, he expects his investment to grow to $21,000, but it can also go down to $9,000 if the economy deteriorates. If the economy stays the same, his investment will stay at $12,000.

a.

What is the expected value of his investment?

Expected value $

b.

What should the investor do if he is risk neutral?

Investor invest the $12,000.

c. Is the decision clear-cut if he is risk averse?

Yes

No

rev: 08_07_2013_QC_33420, 11_01_2013_QC_37895

3.

Investment advisors recommend risk reduction through international diversification. International investing allows you to take advantage of the potential for growth in foreign economies, particularly in emerging markets. Janice Wong is considering investment in either Europe or Asia. She has studied these markets and believes that both markets will be influenced by the U.S. economy, which has a 18% chance for being good, a 55% chance for being fair, and a 27% chance for being poor. Probability distributions of the returns for these markets are given in the accompanying table.

State of the

U.S. Economy Returns

in Europe Returns

in Asia

Good 19% 16%

Fair 9% 6%

Poor −8% −7%

a.

Find the expected value and the standard deviation of returns in Europe and Asia. (Round your intermediate calculations to 4 decimal places and final answers to 2 decimal places.)

Europe Asia

Expected value % %

Standard deviation

b. What will Janice pick as an investment if she is risk neutral?

Investment in Asia

Investment in Europe

rev: 08_07_2013_QC_33420, 03_03_2015_QC_CS-8335

4.

A basketball player is fouled while attempting to make a basket and receives two free throws. The opposing coach believes there is a 30% chance that the player will miss both shots, a 16% chance that he will make one of the shots, and a 54% chance that he will make both shots.

a.

Construct the appropriate probability distribution. (Round your answers to 2 decimal places.)

x P(X = x)

0

1

2

b.

What is the probability that he makes no more than one of the shots? (Round your answer to 2 decimal places.)

Probability

c.

What is the probability that he makes at least one of the shots? (Round your answer to 2 decimal places.)

Probability

rev: 09_13_2013_QC_35141

5.

(Use computer) A committee of 68 members consists of 51 men and 17 women. A subcommittee consisting of 8 randomly selected members will be formed.

a.

What are the expected number of men and women in the subcommittee?

Expected

Number

Men

Women

b.

What is the probability that at least four of the members in the subcommittee will be women? (Round your answer to 4 decimal places.)

Probability

6.

A professor has learned that nine students in her class of 29 will cheat on the exam. She decides to focus her attention on twelve randomly chosen students during the exam.

a.

What is the probability that she finds at least one of the students cheating? (Round your intermediate calculations and final answers to 4 decimal places.)

Probability

b.

What is the probability that she finds at least one of the students cheating if she focuses on fourteen randomly chosen students? (Round your intermediate calculations and final answers to 4 decimal places.)

Probability

rev: 08_07_2013_QC_33420

7.

(Use computer) Assume that X is a Poisson random variable with μ = 24. Calculate the following probabilities. (Round your intermediate calculations and final answers to 4 decimal places.)

a. P(X ≤ 15)

b. P(X = 18)

c. P(X > 20)

d. P(23 ≤ X ≤ 31)

rev: 08_07_2013_QC_33420

8.

Assume that X is a binomial random variable with n = 4 and p = 0.68. Calculate the following probabilities. (Round your intermediate and final answers to 4 decimal places.)

a. P(X = 3)

b. P(X = 2)

c. P(X ≥ 2)

rev: 04_26_2013_QC_29765; rev: 08_07_201

9.

Professor Sanchez has been teaching Principles of Economics for over 25 years. He uses the following scale for grading.

Grade Numerical Score Probability

A 4 0.120

B 3 0.210

C 2 0.400

D 1 0.155

F 0 0.115

Part (a) omitted

b.

Convert the above probability distribution to a cumulative probability distribution. (Round your answers to 3 decimal places.)

Grade P(X ≤ x)

F

D

C

B

A

c.

What is the probability of earning at least a B in Professor Sanchez’s course? (Round your answer to 3 decimal places.)

Probability

d.

What is the probability of passing Professor Sanchez’s course? (Round your answer to 3 decimal places.)

Probability

rev: 02_28_2014_QC_45290

10.

(Use computer) Assume that X is a hypergeometric random variable with N = 48, S = 22, and n = 12. Calculate the following probabilities. (Round your answers to 4 decimal places.)

a. P(X = 10)

b. P(X ≥ 2)

c. P(X ≤ 5)

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