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)