Events that are independent of themselves
Another interesting bit of probability trivia came up today in a discussion of random versus deterministic encryption. In particular, it's possible to have an event that's independent of itself. Here's why.
P(A ∩ B) = P(A) P(B)
So an event A is independent of itself if we have
P(A ∩ A) = P(A) P(A)
P(A) = P(A)2
That's true if either P(A) = 0 or P(A) = 1, in which case the event A is independent of itself.