Prove that for every two events A and B, the probability that exactly one of the events will occur is given by: P(A) + P(B) - 2P(A ∩ B)

When we are determining the probability that only of two
events A and B occurs, we do not want to include the case when both the events occur.
The value of the probability of A occurring P(A), includes the probability of A and B
occurring too. Similarly the probability of the event B occurring P(B), includes the
probability of the events B and A occurring too.

We see
that when we add P(A) and P(B) we are adding the probability that both A
and B occur twice. It is therefore required to
eliminate this part from the final result that we obtain. This is the reason behind
subtracting 2*P(A and B) from P(A) + P(B).

As
the probability that exactly one of two events A and B occur should not include the
probability of both the events occurring, the required probability is P(A) + P(B) -
2*P(A and B)