Addition Theorem on Probability

For any two event A, B the Probability of A union B equals to probability of A added to probability of B minus probability of A intersection B.

Proof :-

Suppose, the simple events in A∩B are γ₁, γ₂, γ₃,…………,γ_k

simple events in A are
α₁, α₂, α₃,……….,α_m, γ₁, γ₂, γ₃,…………,γ_k

simple events in B are β₁, β₂, β₃,……….,β_n, γ₁, γ₂, γ₃,…………,γ_k

Then simple events in A∪B = α₁, α₂, α₃,……….,α_m, β₁, β₂, β₃,……….,β_n, γ₁, γ₂, γ₃,…………,γ_k

Number of simple events in A∩B = k
Number of simple events in A = m+k
Number of simple events in B = n+k
Number of simple events in A∪B = m+n+k

If we suppose number of simple events in S = N, then

considering L.H.S(left hand side of equation above)
P(A) + P(B) – P(A∩B)
= (m+k)/N + (n+k)/N – k/N
= (m+n+k)/N
= P (A∪B)

Note that, if A and B are exclusive events then P (A∪B) = P(A) + P(B)

Remark:- For any three events A,B,C
P(A∪B∪C) = P(A) + P(B) + P(C) – P(A∩B) – P(B∩C) – P(C∩A) + P(A∩B∩C)

And

1. If A,B,C are mutually exclusive events, then P(A∪B∪C) = P(A) + P(B) + P(C)
2. If A,B,C are mutually exclusive and exhaustive events, then P(A∪B∪C) = P(A) + P(B) + P(C) = 1

In general, if we consider A₁, A₂, A₃,………,A_n are any n elements then

Note :-

1. If A₁, A₂, A₃,……..,A_n are any n mutually exclusive events, then
   
2. If A₁, A₂, A₃,……..,A_n are mutually exclusive and exhaustive events then
   

Since A and complimentary element of A are exclusive and exhaustive for any event A we get,

If A,B are two events such that A is subset of B then P(B-A) = P(B) – P(A)
Note that, for any two events A,B P(B-A) = P(B) – P(A∩B)