Homework 1 Solution Stat 134, Sly, Fall 2015
1.3.11 Write A [ B [ C = (A [ B) [ C and use the inclusion{exclusion formula three times
to derive the inclusion{exclusion formula for 3 events:
P(A [ B [ C) = P(A) + P(B) + P(C) - P(AB) - P(AC) - P(BC) + P(ABC)
Answer By inclusion-exclusion for n = 2,
P(A [ B [ C) = P[(A [ B) [ C] = P(A [ B) + P(C) - P[(A [ B)C]
= P(A) + P(B) - P(AB) + P(C) - P[(A [ B)C]:
Now (A [ B)C = (AC) [ (BC), which is clear from a Venn diagram. So by another
application of inclusion-exclusion for n = 2,
P[(A [ B)C] = P[(AC) [ (BC)] = P(AC) + P(BC) - P[(AC)(BC)]
= P(AC) + P(BC) - P(ABC)