MA 2631 Probability Theory Assignment 3

MA 2631
Probability Theory
Assignment 3
due on Friday, September 10
based on Lectures of Chapter 2.1–2.4
1. Let Ω be a sample space, P a probability and E, F events. Prove that
P[E
c ∪ F
c
] ≤ 2 −

P[E] + P[F]


.
2. A person picks 13 cards out of a standard deck of 52.
i) What is the probability that he has at least one of the four aces in his hand?
ii) What is the probability that he has exactly one ace in his hand?
3. You repeatedly toss a coin until you get a heads. What is the probability that you get
the head on an even-numbered toss?
4. An urn contains twelve balls, four of which are white, three green and five black.
a) We draw three balls, what is the probability that all three balls are of different
color?
b) We draw three balls consecutively, after each drawing placing back the ball in the
earn before drawing the next, what is the probability that the color of all three
drawn balls is different
2
5. Consider the lottery “Mega Millions” (a sample slip you may find on page 4 and 5 – if
you do not know about lotteries, please read the instructions carefully).
i) For one play, what is the probability that you will win the jackpot?
ii) For one play, what is the probability that you have three winning numbers plus the
MEGABALL?
6. 25 WPI math majors are joining their classes. 14 go to Probability, 12 go to Linear
Programming and 9 to Discrete Mathematics. 7 take both, Probability and Linear
Programming, 4 both Probability and Discrete Mathematics and 5 Linear Programming
and Discrete Mathematics and 3 all three classes.
a) If a student is picked at random, what is the probability that she is not any of the
three classes?
b) If a student is chosen randomly, what is the chance that he takes exactly one of
these three classes?
c) If two students are chosen randomly, what is the chance that at least one is taken
any of these three classes?
8 points per problems
Additional practice problems (purely voluntary - no points, no credit, no
grading):
Standard Carlton and Devore, Section 1.2: Exercises 14, 15, 16, 19, 20, 22, 25, 27, 28, 30
Hard Show that one cannot define a probability such that the set of natural numbers N forms
a Laplacian sample space (i.e., all outcomes/natural numbers are equally likely)
3
4