Bonferroni Inequality

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

$$ \begin{eqnarray*} & &P(A_1\cup A_2)= P(A_1)+P(A_2) -P(A_1\cap A_2)\leq 1 \\ \implies & & P(A_1\cap A_2) \geq P(A_1) +P(A_2) -1. \end{eqnarray*} $$

Hence from above inequality, it is clear that the inequality \eqref{bof} is true for $n=2$.

Suppose the inequality \eqref{bof} is true for $n=r$.

$$ \begin{equation}\label{bof2} P\big(\cap_{i=1}^r A_i\big)\geq \sum_{i=1}^r P(A_i) -(r-1). \end{equation} $$

Then

$$ \begin{eqnarray*} P\big(\cap_{i=1}^{r+1} A_i\big)& = & P\big(\cap_{i=1}^r A_i \cap A_{r+1}\big)\\ & \geq & P\big(\cap_{i=1}^r A_i\big) + P( A_{r+1}) -1\\ &\geq & \sum_{i=1}^r P(A_i) -(r-1)+ P(A_{r+1}) -1\\ & & \text{ (Using inequality \eqref{bof2})}\\ & \geq &\sum_{i=1}^{r+1} P(A_i) -(r+1-1). \end{eqnarray*} $$

Hence inequality \eqref{bof} is true for $n=r+1$. So by mathematical induction, the inequality \eqref{bof} is true for all $n$.

Bonferroni's Inequality Example

If $P(A) = 0.9$ and $P(B) = 0.8$, show that $P(A \cap B)\geq 0.7$.

Solution

As $A\cup B \subseteq S$, we have $P(A\cup B) \leq P(S) = 1$.

But

$$ \begin{eqnarray*} P(A\cup B) &\leq& 1\\ P(A) + P(B)- P(A\cap B)& \leq & 1\\ P(A\cap B) &\geq & P(A) +P(B) -1 \end{eqnarray*} $$

Thus

$$ \begin{eqnarray*} P(A\cap B) &\geq & P(A) +P(B) -1\\ P(A\cap B) & \geq & 0.9 +0.8 -1\\ &\geq &0.7. \end{eqnarray*} $$

Conclusion

In this tutorial, you learned about Bonferroni's Inequality and how to prove it.

To read more about the tutorials on Probability Theory refer the link Probability Theory. These tutorials will help you to understand basic concepts of probability and various important results of probability theory along with some numerical solved examples on probability theory.

To learn more about other probability distributions, please refer to the following tutorial:

Probability distributions

Let me know in the comments if you have any questions on Bonferroni's Inequality and your thought on this article.

VRCBuzz co-founder and passionate about making every day the greatest day of life. Raju is nerd at heart with a background in Statistics. Raju looks after overseeing day to day operations as well as focusing on strategic planning and growth of VRCBuzz products and services. Raju has more than 25 years of experience in Teaching fields. He gain energy by helping people to reach their goal and motivate to align to their passion. Raju holds a Ph.D. degree in Statistics. Raju loves to spend his leisure time on reading and implementing AI and machine learning concepts using statistical models.

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