Math 477 Homework 1 Solutions
Chapter 1
10. We roll a fair die repeatedly until we see the number four appear and then we stop. The
outcome of the experiment is the number of rolls.
(a) Following Example 1.16 describe a sample space Ω and a probability measure P to
model this situation.
(b) Calculate the probability that the number four never appears.
12. We roll a fair die repeatedly until we see the number four appear and then stop.
(a) What is the probability that we need at most 3 rolls?
(b) What is the probability that we need an even number of die rolls?
14. Assume that P (A) = 0.4 and P (B) = 0.7. Making no further assumptions on A and B,
show that P (AB) satisfies 0.1 ≤ P (AB) ≤ 0.4.
28. We have an urn with m green balls and n yellow balls. Two balls are drawn at random.
What is the probability that the two balls have the same color?
(a) Assume that the balls are sampled without replacement.
(b) Assume that the balls are sampled with replacement.
(c) When is the answer to part (b) larger than the answer to part (a)? Justify your
answer. Can you give an intuitive explanation for what the calculation tells you?
36. (a) Let (X, Y ) denote a uniformly chosen random point inside the unit square
[0, 1]2 = [0, 1] × [0, 1] = {(x, y) : 0 ≤ x, y ≤ 1}.
Let 0 ≤ a ≤ b ≤ 1. Find the probability P (a < X < b), that is, the probability that
the x-coordinate X of the chosen point lies in the interval (a, b).
(b) What is the probability P (|X − Y | ≤ 1/4)?
41. Imagine there is a game of three players where exactly one player wins in the end and all
players have equal chances of being the winner. The game is repeated four times. Find
the probability that there is at least one person who wins no games.
Hint. Let Ai = {person i wins no games} and utilize the inclusion-exclusion formula.
42. Suppose P (A) > 0.8 and P (B) > 0.5.
(a) From the textbook: Show that P (AB) > 0.3.
(b) Add-on: What are the possible values for P (Ac B)?
52. (Challenge, graded for effort only) Three married couples (6 guests altogether) attend a
dinner party. They sit at a round table randomly in such a way that each outcome is
equally likely. What is the probability that somebody sits next to his or her spouse?
Hint. Label the seats, the individuals, and the couples. There are 6! = 720 seating arrangements altogether. Apply inclusion-exclusion to the events Ai = {the ith couple sit next to
each other}, i = 1, 2, 3. Count carefully the numbers of arrangements in the intersections
of the Ai s.