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 = λ_{1}, λ_{2},……..,λ_{k}

Simple events in A = α_{1}, α_{2},……,α_{m},λ_{1}, λ_{2},……..,λ_{k}

Simple events in B = β_{1}, β_{2},……,β_{n}, λ_{1}, λ_{2},……..,λ_{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

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)