## Bonferroni Inequality

Bonferroni’s Inequality For $n$ events $A_1,A_2,\cdots, A_n$ $$$$\label{bof} P\big(\cap_{i=1}^n A_i\big)\geq \sum_{i=1}^n P(A_i) -(n-1).$$$$ Proof For $n =2$, $$\begin{equation*} P(A_1\cup A_2)= P(A_1)+P(A_2) -P(A_1\cap A_2) \end{equation*}$$ But $P(A_1\cup A_2)\leq 1$. Using this in above equation, we have  \begin{eqnarray*} & &P(A_1\cup A_2)= P(A_1)+P(A_2) -P(A_1\cap A_2)\leq 1 \\ \implies & & P(A_1\cap A_2) …