Mathematics 426
Robert Gross
Homework 1
Answers
1. The math department is choosing a schedule of 5 speakers, 1 each for the next 5 weeks. Possible
possible speakers are Alice, Bob, Carol, Dave, Ed, Felix, George, Howie, Indira, and Janice. Assume
no one s

Math 4426: Fall 2016
Exam 1: Solutions
Page 1 of 5
1. (15 pts): Let A and B be two events with P(A) = 0.3 and P(B) = 0.8. You do NOT need to
explain your answers for this question.
(a) What is the largest possible value for P(A B)?
(b) What is the smalles

Math 4426
Page 1 of 4
Fall 2016
Homework # 3 Solutions
Due date: Friday 9/23 in class
Problem 3.6:
We are given an urn with 8 white balls and 4 black balls, from which we will select 4 balls
at random. Let A be the event that the first and third balls dra

Math 4426
Page 1 of 5
Fall 2016
Homework # 2 Solutions
Due date: Wednesday 9/14 in class
Problem 2.8:
(a) Since A B = , we have that P(A B) = P(A) + P(B) = 0.8.
(b) Since A B = , it follows that A B c . Hence, P(A B c ) = P(A) = 0.3.
(c) P(A B) = P() = 0.

Math 4426
Page 1 of 3
Fall 2016
Homework # 1 Solutions
Due date: Wednesday 9/7 in class
Problem 1.10:
(a) 8! = 40,320
(b) A and B next to one another: 2 7 6! = 10,080
2 ways for A sitting next to B, 7 positions in the row, 6! permutations for the
others.

Mathematics 426
Robert Gross
Homework 3
Answers
1. Suppose that A, B, and C are events, and you know that
P(A) = 0.72
P(A B) = 0.78
P(B) = 0.50
P(B C) = 0.84
P(C) = 0.64
P(A B C) = 0.95
P(A C) = 0.91
What is P(ABC)?
Answer : We compute
P(AB) = P(A) + P(B)

Mathematics 426
Robert Gross
Homework 6
Answers
1. Find events E, F, and G so that
P(E|G) > P(E)
P(F|G) > P(F)
P(EF|G) < P(EF)
Hint : One way to do this is to let the experiment consist of rolling a red die and a green die. Let
E be the probability that t

Mathematics 426
Robert Gross
Homework 5
Answers
1. Suppose that I roll 9 identical ordinary (cubical) dice. What is the probability that all
6 numbers from 1 to 6 are visible on the 9 dice?
Answer : This should be done using the Inclusioncfw_Exclusion Pri

Mathematics 426
Robert Gross
Homework 7
Answers
1. Suppose that X is a Poisson random variable with parameter . Show that E[Xn ] =
E[(X + 1)n1 ]. Use this result to compute E[X], E[X2 ], and E[X3 ].
Answer : We have
E[Xn ] =
kn P ( X = k) =
k=0
=
kn e
k=0

Mathematics 426
Robert Gross
Homework 10
Answers
1. Suppose that X1 and X2 are independent discrete random variables with distribution
P(X1 = 0) = P(X2 = 0) = P(X1 = 1) = P(X2 = 1) = 1 . Let X = X1 +X2 .
2
2
(a ) What is the distribution for X?
(b ) Show

Mathematics 426
Robert Gross
Homework 8
Answers
1. Suppose that X is a discrete random variable which takes on integer values. Show that
P(X > k) P(X < k).
E[X] =
k=0
Answer : We know that E[X] =
k kP (X
sum, we can continue:
1
E[X] =
kP(X = k) +
k=
=
= k

Mathematics 426
Robert Gross
Homework 9
Answers
1. Suppose that X is a normal random variable with mean and standard deviation .
Suppose that P(X > 9) = P(X < 3) = 0.1. Compute and as accurately as possible.
Answer : Fortunately, symmetry helps out in thi

Mathematics 426
Robert Gross
Homework 2
Answers
1. Two dierent cards are selected at random from an ordinary deck of 52 playing cards. What is
the probability that they form a blackjack? A blackjack occurs when one of the cards is an ace,
and the other is

Mathematics 426
Robert Gross
Homework 4
Answers
1. Suppose that I roll 7 identical ordinary (cubical) dice.
(a ) What is the probability of exactly 1 pair of matching numbers on the 7 dice?
(b ) What is the probability of 3 pairs of distinct matching numb

Math 4426
Page 1 of 4
Fall 2016
Homework # 5 Solutions
Due date: Friday 10/14 in class
Problem 4.38:
Let X be a random variable with E(X) = 1 and Var(X) = 5.
(a) Recall that Var(X) = E(X 2 ) [E(X)]2 . Hence, 5 = E(X 2 ) (1)2 , which implies
E(X 2 ) = 6. T