MATH 247 - Probability

This course is concerned with advanced topics of probability theory, not
typically covered by introductory statistics courses, such as Math127ab,
Math218, and Math219. Some of the more in-depth subjects to be dealt with
are: combinatorics (the mathematics of counting), probability models
(including various types of distributions for random variables), limit
theorems (weak and strong versions of the law of large numbers), and
stochastic processes (in particular, discrete Markov processes).

Here a few typical questions that can be resolved using the above
mathematical tools:

(1) At a party 20 men take off their hats. The hats are then mixed up and
each man randomly selects one. What is the probability that no man selects
his own hat?

(2) A pipe-smoking mathematician carries, at all times, 2 matchboxes, 1 in
his left-hand pocket and 1 in his right-hand pocket. Each time he needs a
match he is equally likely to take it from either pocket. Consider the moment
when the mathematician first discovers that one of his matchboxes is empty.
If it is assumed that both matchboxes initially contained 30 matches, what is
the probability that there are at least 10 matches in the other box?

(3) A person goes for a run each morning. When he leaves his house for his
run he is equally likely to go out either the front or the back door;
similarly, when he returns he is equally likely to go to either the front or
the back door. The runner owns 5 pairs of running shoes which he takes off
after the run at whichever door he happens to be at. If there are no shoes at
the door from which he leaves to go running he runs barefooted. What
proportion of time does he run barefooted?

(4) Five hundred voters are selected at random and asked who they voted for. 
What is the probability that the candidate whom the majority of those
surveyed named won the election?