Engineering
Mathematics
Addition Theorem on Probability
Sets
Basic Definitions Related to Probability
Question
If A,B and C are three events, then
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Solution
We have
P(exactly two of A, B, C occur)
= P(A⋂B) + P(B⋂C) + P(C⋂A) – 3P(A⋂B⋂C) ≤ P(A⋂B) + P(B⋂C) + P(C⋂A)
Also P(AUBUC) ≤ P(AUB) + P(C) ≤ P(A) + P(B) + P(C)
Next P (exactly one of A, B, C occurs)
= P(A) + P(B) + P(C) – 2P(A⋂B) – 2P(B⋂C) – 2P(C⋂A) + 3P(A⋂B⋂C)
= P(A) + P(B) + P(C) – P(A⋂B) – P(B⋂C) – P(C⋂A) – [P(A⋂B) + P(B⋂C) + P(C⋂A) – 3P(A⋂B⋂C)]
= P(A) + P(B) + P(C) – P(A⋂B) – P(B⋂C) – P(C⋂A) – P (exactly two of A, B, C occur) ≤ P(A) + P(B) + P(C) – P(A⋂B) – P(B⋂C) – P(C⋂A)
Lastly P (A and at least one of B, C occur)
= P[(A⋂B⋂C)] = P[(A⋂B) U (A⋂C)]
= P(A⋂B) + P(A⋂C) – P[(A⋂B) ⋂ (A⋂C)]
= P(A⋂B) + P(A⋂C) – P(A⋂B⋂C) ≤ P(A⋂B) + P(A⋂B) + P(A⋂C).