Probability

Let $A$ and $B$ be events with probabilities $\mathbb {P}\left(A\right) = \frac{3}{4}$ and $\mathbb {P}\left(B\right) = \frac{1}{3}$. Show that $\frac{1}{12} \leq \mathbb {P}\left(A \cap B\right) \leq \frac{1}{3}$, and give examples to show that both extremes are possible. Find corresponding bounds for $\mathbb {P}\left(A \cup B\right)$.

A fair coin is tossed repeatedly. Show that, with probability one, a head turns up sooner or later. Show similarly that any given finite sequence of heads and tails occurs eventually with probability one. Explain the connection with Murphy's Law.

Six cups and saucers come in pairs: there are two cups and saucers which are red, two white, and two with stars on. If the cups are placed randomly onto the saucers (one each), find the probability that no cup is upon a saucer of the same pattern.

Let $A_{1}, A_{2}, \ldots , A_{n}$ be events where $n \geq 2$, and prove that

In each packet of Corn Flakes may be found a plastic bust of one of the last five Vice-Chancellors of Cambridge University, the probability that any given packet contains any specific Vice-Chancellor being $\frac{1}{5}$, independently of all other packets. Show that the probability that each of the last three Vice-Chancellors is obtained in a bulk purchase of six packets is $1-3\left(\frac{4}{5}\right)^{6}+3\left(\frac{3}{5}\right)^{6}-\left(\frac{2}{5}\right)^{6}$.

Let $A_{r}, r \geq 1$, be events such that $\mathbb {P}\left(A_{r}\right) = 1$ for all $r$. Show that $\mathbb {P}\left(\bigcap_{r = 1}^{\infty } A_{r}\right) = 1$.

You are given that at least one of the events $A_{r}, 1 \leq r \leq n$, is certain to occur, but certainly no more than two occur. If $\mathbb {P}\left(A_{r}\right) = p$, and $\mathbb {P}\left(A_{r} \cap A_{s}\right) = q, r \neq s$, show that $p \geq 1 / n$ and $q \leq 2 / n$.

You are given that at least one, but no more than three, of the events $A_{r}, 1 \leq r \leq n$, occur, where $n \geq 3$. The probability of at least two occurring is $\frac{1}{2}$. If $\mathbb {P}\left(A_{r}\right) = p, \mathbb {P}\left(A_{r} \cap A_{s}\right) = q, r \neq s$, and $\mathbb {P}\left(A_{r} \cap A_{s} \cap A_{t}\right) = x, r < s < t$, show that $p \geq 3 /(2 n)$, and $q \leq 4 / n$.