theory of probability 2

MATH 464

HOMEWORK 3

SPRING 2013

The following assignment is to be turned in on Thursday, February 7, 2013.

1. Three couples are invited to a dinner party. They will independently show up with probabilities 0.9, 0.8, and 0.75 respectively. Let N be the number of couples that show up. Calculate the probability that N = 2

2. Statistics show that 5% of men are color blind and 0.25% of women are color blind. If a person is randomly selected from a room with 35 men and 65 women, what is the likelihood that they are color blind?

3. Do Exercise 26 on page 14 of the book.

4. On a multiple choice exam with four choices for each question, a student either knows the answer to a question or marks it at random. Suppose the student knows the answers to 60% of the exam questions. If he marks the answer to question 1 correctly, what is the probability that he knows the answer to that question?

5. In a certain city, 30% of the people are conservative, 50% are liberals, and 20% are independents. In a given election, 2/3 of the conservatives voted, 80% of the liberals voted, and 50% of the independents voted. If we pick a voter at random, what is the probability that this person is a liberal?

6. Let (Ω,F, P ) be a probability space and suppose that {An}∞n=1 is an increasing sequence of events. For each integer n ≥ 1, set

Cn =

{ A1 if n = 1 An \An−1 for n ≥ 2.

Show that the Cn’s are mutually disjoint and that

∞⋃ n=1

An = ∞⋃ n=1

Cn .

1

2 SPRING 2013

7. Let (Ω,F, P ) be a probability space and suppose that {An}∞n=1 is a sequence of events. Set

Bn = ∞⋃

m=n

Am and Cn = ∞⋂

m=n

Am

It is clear that Bn is a decreasing sequence of events, while Cn is an increasing sequence of events. Show that

B = ∞⋂ n=1

Bn = {ω ∈ Ω : ω ∈ An for infinitely many values of n}

and

C = ∞⋃ n=1

Cn = {ω ∈ Ω : ω ∈ An for all but finitely many values of n}

8. Do exercise 4 on page 24 of the book.

9. Suppose we roll two fair 6-sided dice. Let X be a random variable corresponding to the minimum value of the two rolls. Find the probability mass function fX corresponding to the random variable as a table of values (see below).

10. The probability mass function of a discrete random variable X is given below as a table of values. Compute the following: a) the probability that X is even (here we regard 0 and -4 as even) b) the probability that 1 ≤ X ≤ 8 c) the probability that X is -4 given that X ≤ 0 d) the probability that X ≥ 3 given that X > 0

x -4 -1 0 2 4 5 6 fX(x) 0.15 0.2 0.1 0.1 0.2 0.2 0.05