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?