Define the indicator function $I_{A}$ of an event $A$.

Let $I_{i}$ be the indicator function of the event $A_{i}, 1 \leq i \leq n$, and let $N=\sum_{1}^{n} I_{i}$ be the number of values of $i$ such that $A_{i}$ occurs. Show that $E(N)=\sum_{i} p_{i}$ where $p_{i}=P\left(A_{i}\right)$, and find $\operatorname{var}(N)$ in terms of the quantities $p_{i j}=P\left(A_{i} \cap A_{j}\right)$.

Using Chebyshev's inequality or otherwise, show that

$$P(N=0) \leq \frac{\operatorname{var}(N)}{\{E(N)\}^{2}}$$