Question

Q1:

a) Re-derive the inclusion-exclusion principle for two events using only the probability axioms.

Probability axioms: Given an event A in Ω:

A1) P(A) >= 0

A2) P(Ω) = 1

A3) P(U (from i=1 to n) A_i) = Σ (from i=1 to n) P(A_i) - if A_i's are disjoint/ mutually exclusive

Inclusion Exclusion Principle for two events: (A U B) = (A) + (B) + (A ∩ B)

b) Then, using only the axioms and the inclusion-exclusion principle for two events, derive the formula for the inclusion-exclusion principle for three events.

Inclusion Exclusion Principle for three events: (A U B U C) = (A) + (B) + (C) - (A ∩ B) - (A ∩ C) - (B ∩ C)

Answer 1