(a) Let ℙ(A) denote the probability of the occurrence of an...

(a) Let $\mathbb{P}(A)$ denote the probability of the occurrence of an event $A$. Prove carefully, for events $A_{1}, A_{2}, \ldots, A_{n}$, that $$ \begin{aligned} \mathbb{P}\left(\bigcup_{i=1}^{n} A_{i}\right)=\sum_{i} \mathbb{P}\left(A_{i}\right) &-\sum_{i<j} \mathbb{P}\left(A_{i} \cap A_{j}\right)+\sum_{i<j<k} \mathbb{P}\left(A_{i} \cap A_{j} \cap A_{k}\right)-\cdots \\ &+(-1)^{n+1} \mathbb{P}\left(\bigcap_{i} A_{i}\right) \end{aligned} $$ (b) One evening, a bemused lodge-porter tried to hang $n$ keys on their $n$ hooks, but only managed to hang them independently and at random. There was no limit to the number of keys which could be hung on any hook. Otherwise, or by using (a), find an expression for the probability that at least one key was hung on its own hook. The following morning, the porter was rebuked by the Bursar, so that in the evening she was careful to hang only one key on each hook. But she still only managed to hang them independently and at random. Find an expression for the probability that no key was then hung on its own hook. Find the limits of both expressions as $n$ tends to infinity. You may assume that, for real $x$,

(a) Let $\mathbb{P}(A)$ denote the probability of the occurrence of an event $A$. Prove carefully, for events $A_{1}, A_{2}, \ldots, A_{n}$, that
$$
\begin{aligned}
\mathbb{P}\left(\bigcup_{i=1}^{n} A_{i}\right)=\sum_{i} \mathbb{P}\left(A_{i}\right) &-\sum_{i<j} \mathbb{P}\left(A_{i} \cap A_{j}\right)+\sum_{i<j<k} \mathbb{P}\left(A_{i} \cap A_{j} \cap A_{k}\right)-\cdots \\
&+(-1)^{n+1} \mathbb{P}\left(\bigcap_{i} A_{i}\right)
\end{aligned}
$$
(b) One evening, a bemused lodge-porter tried to hang $n$ keys on their $n$ hooks, but only managed to hang them independently and at random. There was no limit to the number of keys which could be hung on any hook. Otherwise, or by using (a), find an expression for the probability that at least one key was hung on its own hook.
The following morning, the porter was rebuked by the Bursar, so that in the evening she was careful to hang only one key on each hook. But she still only managed to hang them independently and at random. Find an expression for the probability that no key was then hung on its own hook.
Find the limits of both expressions as $n$ tends to infinity.
You may assume that, for real $x$,

Key Concepts

Asymptotic Analysis in Probability

This area focuses on the behavior of probability expressions and combinatorial formulas as the number of elements grows indefinitely. It is crucial for understanding limits, such as the convergence of the probability of events (like no fixed points in a permutation) to well-known constants, and provides insight into the long-term trends and approximations useful in both theoretical and applied probability.

Inclusion-Exclusion Principle

This is a fundamental principle in probability and combinatorics used to calculate the probability (or cardinality) of a union of events by alternately adding and subtracting the probabilities (or counts) of intersections of these events. It corrects for overcounting when multiple events overlap, and forms the basis of many arguments for systems where events are not mutually exclusive.

Derangements

A derangement is a permutation where none of the objects appears in its designated or original place. In combinatorial probability, this concept is key when determining the likelihood that no item is fixed under a random permutation. It involves an alternating sum that mirrors the inclusion-exclusion principle and is noted for its asymptotic behavior approaching 1/e as the number of objects increases.

Combinatorial Counting and Random Permutations

This concept encompasses the methods for counting the number of ways objects can be arranged, particularly when subject to specific restrictions such as fixed points or the possibility of multiple assignments. In probability, it provides a framework for comparing the total number of outcomes to the number of favorable cases, thereby determining probabilities in scenarios involving random independent assignments or unique placements.

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