EE 325: Probability and Random Processes
Problem Set 2
1. Prove that
(a) P(A ∩ B) ≥ P(A) + P(B) − 1
(b) P(A1 ∩ A2 ∩ · · · ∩ An ) ≥ P(A1 ) + P(A2 ) + · · · + P(An ) − (n − 1).
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2. Let A, B be events such that P(A) = 4
and P(B) = 21 .
(a) Find maximum possible values for P(A ∪ B) & P(A ∩ B)
(b) Find values of P(A ∪ B) & P(A ∩ B) if A, B are independent.
3. Let Ω = {1, 2, 3, ..., p} where p is a prime, F be the collection of all subsets of Ω, and
P(A) = |A|
p
∀A ∈ F. Show that, if A and B are independent events, then at least one of
A and B is either ϕ or Ω.
4. A wise man presents you with an urn containing an equal number of red and black
marbles. If you reach in and draw two marbles (uniformly and independently) at random
from the urn, the probability that they’re of matching color is necessarily less than 12 .
Here’s the tricky part: if the colors of those first two marbles do match, and you reach in
to grab two more, the probability of those next two being of matching color is exactly 12 .
How can that be? What is the total number of balls in the urn - red plus black - before
you take any out?
5. You invite some friends to a hat-themed party. As the night progresses, you decide to
play a game: everyone will line up and place their hat on the ground in front of them.
The guests will then be reordered randomly, with all orderings equally likely. Afterward,
each guest will grab the hat in front of them. We’re interested in the probability that at
least one guest is wearing their own hat at the end of the game. What does that probability
converge to as the number of guests grows infinitely large?
S for each k, we have Ak ⊆ Ak+1 , and that
6. Let A, B, A1 , A2 , ... be events. Suppose that
Ak is independent of B, ∀k ≥ 1. If A = k∈N Ak , then show that B is independent of
A.
7. A man is known to speak the truth 3 out of 4 times. He throws a die and reports that it is
a six. Find the probability that it is actually a six.
8. A box contains a white balls and b black balls. Balls are drawn at random one at a time
without replacement. Find the probability of encountering a white ball in the k th draw.
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9. A, B and C are evenly matched tennis players. Initially A and B play a set, and the
winner then plays C. This continues, with the winner always playing the waiting player,
until one of the players has won two sets in a row. That player is then declared the overall
winner. For each player, find the probability of him/her being the overall winner.
10. A three-man jury has two members each of whom independently has probability p of
making the correct decision and a third member who flips a coin for each decision (ma-
jority rules). A one-man jury has probability p of making the correct decision. Which
jury has the better probability of making the correct decision?
