theory of probability 1

MATH 464

HOMEWORK 2

SPRING 2013

The following assignment is to be turned in on Thursday, January 31, 2013.

1. Suppose we pick a letter at random from the word MISSISSIPPI. Write down a sample space and give the probability of each outcome?

2. In a group of students, 25% smoke, 60% drink, and 15% do both. What percentage of the students that either smoke or drink?

3. Let (Ω,F, P ) be a probability space. Suppose A, B ∈F with:

P (A) = 1

3 P (Ac ∩Bc) =

1

2 and P (A∩B) =

1

4 .

What is P (B)?

4. Let (Ω,F, P ) be a probability space. Suppose A, B ∈F with P (A) = 0.4 and P (B) = 0.7. What are the maximum and minimum possible values for P (A∩B)?

5. Let (Ω,F, P ) be a probability space. Let A, B, C ∈F. Prove that

P (A∪B∪C) = P (A)+P (B)+P (C)−P (A∩B)−P (A∩C)−P (B∩C)+P (A∩B∩C) .

This is a special case of a more general result called the inclusion-exculsion formula.

6. Alice and Bob take turns flipping a fair coin. The game is: the first one to heads wins. Bob lets Alice flip first. What is the probability that she wins?

7. An unfair coin has probability 1/3 for heads and 2/3 for tails. Do an experiment where you flip this coin until you get heads and then stop. What is the probability it takes exactly 8 flips, given that it takes at least 6 flips?

8. Let (Ω,F, P ) be a probability space. Let A, B ∈ F with P (B) > 0. Prove that

P (Ac|B) = 1 −P (A|B) . using the definition of conditional probability measures. Do not use the fact that conditional probabilities define probability measures.

1

2 SPRING 2013

9. Let (Ω,F, P ) be a probability space. Suppose A, B ∈F are independent events. Prove that Ac and Bc are independent events.

10. Roll a fair, six-sided die twice. Let A be the event that the first roll is odd. Let B be the event that the second roll is even. Let C be the event that either both rolls are even or both rolls are odd. Show that A, B, and C are pairwise independent, but not independent.