theory of probability 1

  

strict due date.

MATH 464

HOMEWORK

1

SPRING 2013

The following assignment is to be turned in on
Thursday, January 24th, 2013.

1. Let A, B, and C be events (subsets) of a sample space Ω.
Write each of the following events in terms of A, B, and C using intersec-
tions, unions, and complements.

(a) None of A, B, C occurs.
(b) At least two of A, B, C occur.
(c) Exactly one of A, B, C occurs.
(d) Exactly two of A, B, C occur.
(e) Exactly three of A, B, C occur.
Hint: Read Section 1.2 of the book.

2. Do exercises 3, 5, and 6 on pages 5 and 7 of the book.

3. Consider an unfair coin which has probability 1/3 for heads and 2/3 for
tails. Do an experiment where you flip this coin until you get heads and
stop.
(a) Write out a simple sample space.
(b) What is the probability it takes exactly 5 flips?
(c) What is the probability it takes at least 3 flips?

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MATH 464

HOMEWORK 2

SPRING 20

1

3

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.

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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.