Statistical Inference I: J. Lee Assignment 1

Problem 1. An exam has 10 multiple choice questions where each question has 5 possible answers.

- (a) If student goes into exam completely unprepared and guesses on all 10, what is the probability of

getting at least 5 answers right?

- (b) Suppose for each question, the student has probability .60 of knowing the answer, and the student randomly guesses on the remaining questions that he/she does not know. Assuming independence of each question, what is the probability of getting at least 5 answers right?

Problem 2. Suppose that A and B are mutually exclusive events for which P(A) = 0.35 and P(B) = 0.51. What is the probability that

(a) either A or B occurs;

(b) A occurs but B does not;

(c) both A and B occur; (d) neither A nor B occurs?

Problem 3. Consider two events, A and B. You are told that P(A) = 0.75 and that P(B) = 0.81. (a) What is the maximum possible value for P (A ∩ B)? Justify.

(b) What is the minimum possible value for P (A ∩ B)? Justify.

(c) What is the maximum possible value for P (A ∪ B)? Justify.

(d) What is the minimum possible value for P (A ∪ B)? Justify.

Problem 4. A row of 15 cars is stopped at a traffic light. 6 of them are Toyotas, 7 are Hondas, and 2 are Mercedes. Assuming they are ordered randomly, we want to compute the probability that the fifth car in the row is a Mercedes.

- (a) Describe the sample space, S, you would use to solve this problem. Make sure to define any notation you use to describe elements of the sample space.
- (b) Compute the probability that the fifth car is a Mercedes.

Problem 5. A bookstore receives six boxes of books per month on six random days of each month. Suppose that two of those boxes are from one publisher, two from another publisher, and the remaining two from a third publisher. Our goal is to compute the probability that the last two boxes of books received last month are from the same publisher.

- (a) Describe the sample space, S, you would use to solve this problem. Make sure to define any notation you use to describe elements of the sample space.
- (b) Compute the probability that the last two boxes of books received last month are from the same publisher.

Problem 6. Consider an experiment in which two fair dice are rolled. What is the conditional probability that at least one lands on 6, given that the dice land on different numbers?

Problem 7. Suppose that 5 percent of men and 0.25 percent of women are color-blind. A color-blind person is chosen at random. What is the probability of this person being male? Assume that there are an equal number of males and females.

Problem 8. A woman has agreed to participate in an ESP experiment. She is asked to pick, randomly, two distinct integers between 1 and 6 (inclusive).

- (a) What is the probability that the first number she picks is 3 and that the second number is greater than 4?
- (b) What is the probability that both numbers are less than 3?
- (c) What is the probability that both numbers are greater than 3?

Problem 9.

- (a) A gambler has in his pocket two fair coins and one two-headed coin. He selects one of the coins at random; when he flips it, it shows heads. What is the probability that it is a fair coin?
- (b) Suppose that he flips the same coin a second time and again it shows heads. What is now the probability that it is a fair coin?
- (c) Suppose that he flips the same coin a third time and it shows tails. What is now the probability that it is a fair coin?

Problem 10. I tell you that, for two events A and B, P(A) = 0.70,P(B|A) = 0.5, and that A and B are independent. For each statement, say “True”, “False”, or “Can’t Tell”, and give a reason for your answer.

(a) A and B are mutually exclusive (b) A and A ∪ B are independent

(c) P(B)=P(A|B) (d) P(A|B)<P(B|A)

(e) P(B) ≤ P(A)