(a) (i) Suppose that each of three men at a party throws his hat into the center of the
room. The hats are first mixed up and then each man randomly selects a hat.
What is the probability that none of the three men selects his own hat?
(ii) Suppose all n men at a party throw their hats in the center of the room. Each
man then randomly selects a hat. Show that the probability that none of the n
men selects his own hat is
$$ \frac{1}{2!} - \frac{1}{3!} + \frac{1}{4!} - ... - \frac{(-1)^n}{n!} $$
[5+7 marks]
(b) You know that a certain letter is equally likely to be in any one of three different
folders. Let $a_i$ be the probability that you will find your letter upon making a quick
examination of folder i if the letter is, in fact, in folder i, i = 1, 2, 3, (we may have
$a_i$ < 1). Suppose you look in folder 1 and do not find the letter. What is the probability
that the letter is in folder 1?
[5 marks]
(c) In a certain species of rats, black dominates over brown. Suppose that a black rat with
two black parents has a brown sibling.
(i) What is the probability that this rat is a pure black rat (as opposed to being a
hybrid with one black and one brown gene)?
(ii) Suppose that when the black rat is mated with a brown rat, all five of their
offspring are black. Now, what is the probability that the rat is a pure black rat?
[4+4 marks]
