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.

Two events A and B are independent if we have

P(A ∩ B) = P(A) P(B)

So an event A is independent of itself if we have

P(AA) = P(A) P(A)

or

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.

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