Texas A&M University
Department of Mathematics
Volodymyr Nekrashevych
Fall 2011
Math 411 — Problem Set 4
Issued: 09.23
Due: training
4.1. If we flip a coin 5 times, what is the probability that the number of
heads is an even number?
4.2. Three students each have probability 1/3 of solving a problem. What
is the probability that at least one of them will solve the problem?
We assume that the two events (students solving the problem) are
independent.
4.3. Three dice are rolled. For a $1 bet you win $1 for each 6 that appears
(plus your dollar back). If no 6 appears you lose your dollar. What is
the expected value of your win?
4.4. Suppose we roll two tetrahedral dice that have 1, 2, 3, and 4 on their
four sides. Find the distribution for the sum of the three numbers.
4.5. A random variable has P (X = x) = x/15 for x ∈ {1, 2, 3, 4, 5}, and 0
otherwise. Find the mean and variance of X.
4.6. If seven dice are rolled, what is the probability that each of the six
numbers will appear at least once?
4.7. What is the probability of guessing exactly 3 out of 5 multiple choice
questions, if each question has four answers?
4.8. Calls to a toll-free hotline service are made randomly at rate 2 per
minute. The service has five operators, none of whom is currently
busy. Use the Poisson distribution to estimate the probability that in
the next five minute there are less than five calls.
4.9. You pick five cards out of a deck of 52. What is the probability that
you get exactly 2 spades?
4.10. A drawer contains 10 black, 8 brown, and 6 blue socks. If we pick two
socks at random, what is the probability that they match?
4.11. You pick 13 cards out of a deck of 52. What is the probability that
you will not get a card from every suit? (Use the inclusion-exclusion
formula.)