Motivation

Let $A_{1}, \ldots , A_{N}$ be events in a probability space $(\Omega , \mathcal{F}, \mathbb {P})$. For each subset $J$ of $\left\{ 1,2, \ldots , N\right\}$ write $A_{J}$ for $\cap_{i \in J} A_{i}$. Define $S_{k}: = \sum_{\left|J\right| = k} \mathbb {P}\left(A_{J}\right)$, where $\left|J\right|$ denotes the number of indices in $J$. For $0 \leq m \leq N$ show that the probability $\mathbb {P}\left(\left\{ \text{exactly }m\text{ of the }A_i\text{'s occur}\right\} \right)$ equals $\binom {m}{m} S_{m}-\binom {m+1}{m} S_{m+1}+\ldots \pm \binom {N}{m} S_{N}$.

Rederive the assertion of Lemma $\langle 6 \rangle$ by consideration of the net return from the following system of bets:

1. for each $i$, pay $c_{i} p\left(F_{i}\right)$ in order to receive $c_{i}$ if $F_{i}$ occurs, where $c_{i}: = p\left(X \mid F_{i}\right)$;

2. pay $-p(X)$ in order to receive $-X$;

3. for each $i$, make a bet contingent on $F_{i}$, paying $c_{i}$ (if $F_{i}$ occurs) to receive $X$.

Lemma 6: If $\Omega$ is partitioned into disjoint events $F_{1}, \ldots , F_{k}$, and $X$ is a random variable, then $p(X) = \sum_{i = 1}^{k} p\left(F_{i}\right) p\left(X \mid F_{i}\right)$.

For an increasing sequence of events $\left\{ A_{n}: n \in \mathbb {N}\right\}$ with union $A$, show $\mathbb {P}\left(A_{n}\right) \uparrow \mathbb {P}\left(A\right)$.