Boole’s Inequality The Boole’s Inequality Theorem states that "the probability of several events occuring is less than or equal to the sum of the probabilities of each event occuring". For any two events $A$ and $B$, we have $$ \begin{eqnarray*} P(A \cup B) &=& P(A) + P(B) – P(A\cap B)\\ &\leq & P(A) + P(B)\\ …
Bonferroni’s Inequality For $n$ events $A_1,A_2,\cdots, A_n$ $$ \begin{equation}\label{bof} P\big(\cap_{i=1}^n A_i\big)\geq \sum_{i=1}^n P(A_i) -(n-1). \end{equation} $$ Proof For $n =2$, $$ \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) …
