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So I've come across the following inequality for probability measures:

$ P(X \cap Y) \ge P(X) + P(Y) - 1 $

I'm trying to work out why it should be true. I'm sure I'm missing something obvious.

I have the following:

$ P(X \cap Y) = P(X) +P(Y) - P(X \cup Y) \le P(X) +P(Y) - 1 $

This seems to suggest that the inequality is the wrong way round. Have I done something wrong?

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    @MichaelGreinecker because I couldn't accept an answer at the time. There is a minimum time limit before you can accept an answer.2012-05-20

3 Answers 3

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$P(X\cup Y)\leq 1$. Hence $-P(X\cup Y)\geq -1$.

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All the pieces are already on this page in one place or another. Here's how I would put them together: $1 \geq P(X \cup Y) = P(X) + P(Y) - P(X\cap Y)$. So, $1-P(X) - P(Y) \geq -P(X\cap Y)$. Then you use what Michael and Hurkly observed (multiply my negative one on each side and flip the inequality) to find that this implies $P(X) + P(Y) - 1 \leq P(X\cap Y)$, which is what you wanted.

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For what it's worth, I convinced myself of the truth of the inequality as follows: If $X$ and $Y$ are both big ($>0.5$) then they must overlap. So, $P(X)+P(Y)-1$ measures the size of the forced overlap. (obviously it will be less than 0 when no overlap is forced.)