*chevron_left*Probability Theory

# Bonferroni's inequality

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*schedule*Jul 1, 2022

Bonferroni's inequality allows us to define a lower bound for a finite intersection of events.

It states that:

Generalizing to the normal case:

# Proof

From De Morgan's laws we know that:

As the two events are equal, their probabilities will also be equal:

The union bound states that $\mathbb{P}(A_1\,\cup\,A_2) \le \mathbb{P}(A_1) + \mathbb{P}(A_2)$, hence applying it to the above we can rewrite as:

Another way to represent $\mathbb{P}((A_1\,\cap\,A_2)^C)$ is simply $1 -\mathbb{P}(A_1\,\cap\,A_2)$. Doing the likewise for the right hand side also, this gives us:

# Example

Given the following information on sports played by students in a class:

Sport | Number of students | Probability |
---|---|---|

Football only | 2 | 2/25 = 0.08 |

Baseball only | 3 | 3/25 = 0.12 |

Both Football and Baseball | 19 | 19/25 = 0.76 |

Other | 1 | 1/25 = 0.04 |

We can say that:

Students playing Football = "Football only" + "Both Football and Baseball" = 21

Students playing Baseball = "Baseball only" + "Both Football and Baseball" = 22

Hence:

Referring back to Bonferroni's inequality:

We can see that it indeed does hold true here.