Worked examples

There are two roads from A to B and two roads from B to C. Each of the four roads is blocked by snow with probability $p$, independently of the others. Find the probability that there is an open road from A to B given that there is no open route from A to C.

If, in addition, there is a direct road from A to C, this road being blocked with probability $p$ independently of the others, find the required conditional probability.

Calculate the probability that a hand of 13 cards dealt from a normal shuffled pack of 52 contains exactly two kings and one ace. What is the probability that it contains exactly one ace given that it contains exactly two kings?

A symmetric random walk takes place on the integers $0,1,2, \ldots , N$ with absorbing barriers at 0 and $N$, starting at $k$. Show that the probability that the walk is never absorbed is zero.

The so-called 'sure thing principle' asserts that if you prefer $x$ to $y$ given $C$, and also prefer $x$ to $y$ given $C^{\mathrm{c}}$, then you surely prefer $x$ to $y$. Agreed?

A pack contains $m$ cards, labelled $1,2, \ldots , m$. The cards are dealt out in a random order, one by one. Given that the label of the $k$ th card dealt is the largest of the first $k$ cards dealt, what is the probability that it is also the largest in the pack?

A group of $2 b$ friends meet for a bridge soirée. There are $m$ men and $2 b-m$ women where $2 \leq m \leq b$. The group divides into $b$ teams of pairs, formed uniformly at random. What is the probability that no pair comprises 2 men?