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MSIN0022

Mathematics III (Probability Theory)

Assignment 1

 1.  [20 points] You are playing a game of football with your friend, in which you and your friend take turns in shooting the football at the goal. You shoot the football first. The first player to score a goal wins the game.  Each time you shoot, you have a probability p1  of scoring. Each time your friend shoots, she has a probability p2 of shoring. The shots are independent.

(a)  [10 points] What is the probability that you win the game?

(b)  [10 points] What should the values of p1  and p2  be so that you each have 50% chance

of winning the game?

2.  [20 points] Let A, B, and C denote three pairwise independent sets, i.e., A is independent of B , B is independent of C, and A is independent of C . Show that

(a)  [5 points] A and B ∩ C are also independent.

(b)  [15 points] A and B ∪ C are also independent.

3.  [20 points] There are 20 balls in an urn, 5 red balls and 15 black balls.  Suppose that you select 2 balls at random, and let X denote the number of black balls in your sample. What is the probability mass function of X?

4.  [20 points] Assume that a random variable X has the following probability density function (pdf):

'1/3   if 1 ≤ x < 2,

〈

'(0       otherwise,

where a ≥ 0.

(a)  [5 points] What is the value of a?

(b)  [5 points] What is E[X]?

(c)  [5 points] What is Var[X]?

(d)  [5 points] What is P(1 ≤ X ≤ 2.5)?

5.  [7 points] You go to the Engineering Front Building cafeteria for a snack.  You are really hungry, so you decide to buy one chocolate bar, two bags of crisps, and four bags of candy. Assuming that there are 8 options for chocolates, 9 options for crisps, and 10 options for candy, how many possible purchasing combinations do you have?

6.  [6 points] How many different letter arrangements can be made from the letters of the word: “Excitement”?

7.  [7 points] You are a tourist in London, staying for a week. You love to shop while on holiday, so you go to Oxford street, and decide to visit exactly one shop each day.  Assuming that there are 20 shops on Oxford street, how many different possibilities (sequences of 7 different shops) are there?