Question

1. Indicate if each of the following is true or false. If false, provide a counterexample.
(a) The mean of a sample is always the same as the median of it.

(b) The mean of a population is the same as that of a sample.

(c) If a value appears more than half in a sample, then the mode is equal to the value.

2. (Union and intersection of sets) Let Ω = {1,2,...,6}. Suppose each element is equally likely, i.e., P({ω}) = 1/6 for each ω.

(a) Let A be the set of even numbers in Ω and let B be the set of odd numbers in Ω. Obtain A∪B and A∩B.

(b) Let A ⊆ B. What is the probability of P(A\B)?

(c) Let A be the set of even numbers in Ω. Calculate P(Ω\A).

3. (Permutation and combination) There are 10 landmarks in a city. Ann is supposed to stay in the city for 3 days. Because of time constraint, Ann must choose only one landmark for each day. Suppose Ann does not want to visit the same place more than once. How many options Ann should consider?

4. Ann visits a restaurant and finds that there are 5 main dishes and 3 desserts. Ann is considering to choose 2 main dishes and 1 dessert. How many options does she have?

5. Bob is supposed to take an exam. There are 5 questions and the event that each question is correctly answered is 30% and those events are independent. Let X be a random variable for the number of correct answers that Bob will make.

(a) Calculate P(X ≥ 1).

(b) Calculate P(X = 2).

(c) Obtain the cumulative distribution function for X.

(d) Calculate the mean and the variance of X.

6. Ann and Bob play a simple game: each independently draws a number from 1 to 10 each of which is equally likely. The person with a higher number wins. If they draw a same number, then there is a draw.
(a) What is the probability of Ann’s winning before drawing a number?

(b) What is the probability of Ann’s winning conditional on drawing number 7?

(c) Let X1 be a random variable for the number Ann draws; and let X2 be that for Bob. Calculate P(X1 + X2 ≥ 18).

(d) What is the mean of X1 + X2? What is the variance?

7. Ann and Bob are playing a board game. It is almost the end of the game; Ann is supposed to roll a dice first then Bob. Ann wins if the number on the dice is at least 5, while Bob wins if Ann fails to win and the number of his dice tuns out to be at least 3. If both fail then, there will be a draw.
(a) What is the probability of Ann winning the game?

(b) What is the conditional probability of Bob winning the game given that Ann fails to win?

(c) What is the probability of Bob winning the game?

(d) Chris bets on Bob: if Bob wins Chris gets 1, while he loses 2 if Ann wins. A draw gives him 0. What is the mean of the amount of Chris’ winning?

Answer 1