only 3, 6, 7, 8, 9 are needed to be done.

MATH 464

HOMEWORK 3

SPRING 20

1

3

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