P(A∩B) = P(A|B)*P(B) fоr аn event B such thаt P(B)>0. Let's cоnsider а partitiоn E1,...Ek of the sample space, namely E1,...,Ek are mutually exclusive, and E1UE2..UEk =S. suppose probability of each partition is positive so we can define P(A|E1),....,P(A|Ek). P(A∩S) =[a] P(AU (E1UE2..UEk)) = [b] Using distributive property, A∩ (E1UE2..UEk)= (A∩E1)U(A∩E2)U....U(A∩Ek) P( A∩ (E1UE2..UEk)) = P((A∩E1)U(A∩E2)U....U(A∩Ek)) = P(A∩E1)+P(A∩E2)+...+P(A∩Ek). The last equality is using the fact that [c] If E1,...Ek have equal probabilities P(A) =[d]
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