1. Proving Bonferroni's Inequality

Bonferroni's Inequality states that:

$P(A \cap B) \geq P(A) + P(B) - 1$

But the first axiom of probability states that:

$P(\Omega ) = 1$

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Am I allowed to do the following?

$P(A \cap B) \geq P(A \cup B) - 1$

$P(A \cap B) \geq P(A \cup B) - P(\Omega )$

$P(\Omega ) \geq P(A \cup B) - P(A \cap B)$

And the last line must be true because the entire sample size must be at least equal to the union of A and B minus their intersection.

OR

Should i rather try to prove it this way:

Rearrange so that the 1 is isolated...

$1 \geq P(A) + P(B) - P(A \cap B)$

$1 \geq P(A \cup B)$

$P(\Omega ) \geq P(A \cup B)$

Which is also true...

Anyone got an idea which one of these is correct?

EDIT: The second one might be the correct one seeing as A and B are not disjoint. Am i right?

2. $\Omega$ is the sample space. Then $P(A\cup B) \leq P(\Omega) = 1$.

And $P(A \cup B) = P(A) + P(B) - P(A \cap B)$.

Hence $P(A \cap B) \geq P(A) + P(B) -1$.

3. Originally Posted by tukeywilliams
$\Omega$ is the sample space. Then $P(A\cup B) \leq P(\Omega) = 1$.

And $P(A \cup B) = P(A) + P(B) - P(A \cap B)$.

Hence $P(A \cap B) \geq P(A) + P(B) -1$.
Thanks a lot TukeyWilliams +rep+