Boole’s Inequality

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

$$ \begin{eqnarray*} P(A \cup B) &=& P(A) + P(B) - P(A\cap B)\\ &\leq & P(A) + P(B)\\ & & \quad (\because P(A\cap B)\geq 0) \end{eqnarray*} $$

Similarly, for three events $A$, $B$ and $C$, we have

$$ \begin{equation*} P(A \cup B\cup C) \leq P(A) + P(B) +P(C). \end{equation*} $$

Boole's Inequality Theorem

For $n$ events $A_1,A_2,\cdots, A_n$

$$ \begin{equation}\label{bool} P\big(\cup_{i=1}^n A_i\big)\leq \sum_{i=1}^n P(A_i). \end{equation} $$

Proof

For any two events $A_1$ and $A_2$,

$$ \begin{equation}\label{bool01} P(A_1\cup A_2) = P(A_1) + P(A_2) - P(A_1\cap A_2)\leq P(A_1)+P(A_2) \end{equation} $$

Hence \eqref{bool} is true for $n=2$.

Suppose \eqref{bool} is true for $n=r$. That is

$$ \begin{equation}\label{bool2} P\big(\cup_{i=1}^r A_i\big)\leq \sum_{i=1}^r P(A_i). \end{equation} $$

For $n=r+1$,

$$ \begin{eqnarray*} P\big(\cup_{i=1}^{r+1} A_i\big) & = & P\big(\cup_{i=1}^{r} A_i \cup A_{r+1}\big)\\. &\leq & P\big(\cup_{i=1}^{r} A_i\big) + P(A_{r+1}) \text{ (Using \eqref{bool01})}\\ & \leq & \sum_{i=1}^r P(A_i) + P(A_{r+1})\text{ (Using \eqref{bool2})}\\ &\leq & \sum_{i=1}^{r+1} P(A_i). \end{eqnarray*} $$

Hence \eqref{bool} is true for $n = r+1$. Thus by the Principle of mathematical induction, \eqref{bool} is true for all n.

Conclusion

In this tutorial, you learned about Boole's Inequality.

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 Boole'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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