I came across the following problem, and my answer is so close to being correct I wondered where I went wrong:
Let $A, B,$ and $C$ be three events. Show that exactly two of these events will occur with probability $$P(A\cap B) + P(A\cap C) + P(B\cap C) − 3P(A\cap B\cap C)$$
I solved it the following way by using the Inclusion/Exclusion Principle:
$$P\big[(A\cap B)\cup (A\cap C) \cup (B\cap C\big)]=P(A\cap B) + P(A\cap C) + P(B\cap C) − 2P(A\cap B\cap C)$$
I am so close, yet so far, and I can't figure out why, am I not allowed to apply the inclusion/exclusion principle? Did I miscalculate? Or am I not calculating what is asked of me? If so what am I calculating?
ANY help is greatly appreciated :)