Inclusion-exclusion formula for 3 events. write aubuc = (aub) uc and use the inclusion-exclusion formula three times to derive the inclusion-exclusion formula for 3 events:

solution:

We are going to use this property: P (A U B) = P (A) + P (B) - P (AB)

First we treat (A U B) as one event as follows:

P ((A U B) U C) = P (A U B) + P (C) - P ((A U B) C)

= P (A U B) + P (C) - P (AC U BC)

= P (A) + P (B) - P (AB) + P (C) - P (C) - P (AC U BC)

=P (A) + P (B) - P (AB) + P (C) - P (AC) - P (BC) + P (ABC)

= P (A) + P (B) + P (C) - P (AB) - P (AC) - (BC) + P (ABC)