According to Multiplication Theorem of Probability, for any two events A, B probability of A∩B is as given below
P(A∩B) = P(A).P(B/A) = P(B).P(A/B)
Suppose N is the total number of simple events in the sample space S.
Simple events in A∩B = λ1, λ2,……..,λk
Simple events in A = α1, α2,……,αm,λ1, λ2,……..,λk
Simple events in B = β1, β2,……,βn, λ1, λ2,……..,λk
Number of simple events in A∩B = k
Number of simple events in A = m+k
Number of simple events in B = n+k
P(A∩B) = k/N
P(A) = (m+k)/N
P(B/A) = (number of simple events in A∩B) / (total number of simple events in A) = k / (m+k)
from above we get, P(A). P(B/A) = k/N = P(A∩B)
Therefore P(A∩B) = P(A).P(B/A)
Similarly we can prove that P(A∩B) = P(B).P(A/B)
Note :-
- For any three events A,B,C
P(A∩B∩C) = P(A) . P(B/A) . P(C/{A∩B}) - P(B/A) = P(A∩B) / P(A) = k / (m+k)