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Multiplication Theorem on Probability

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Let E and F be two events associated with a sample space S.  Clearly, the set E ∩ F denotes the event that both E and F have occurred. In other words, E ∩ F denotes the simultaneous occurrence of the events E and F. The event E ∩ F is also written as EF. We know that the conditional probability of event E given that F has occurred is denoted by P(E|F) and is given by
P(E|F) = `(P(E ∩ F ))/(P(F)) , P(F) ≠ 0` 
From this result, we can write 
P(E ∩ F) = P(F) . P(E|F)               ... (1) 
Also, we know that 
P(F|E) = `(P(F ∩ E))/(P(E)) , P(E) ≠ 0`

or P(F|E) = `(P(E ∩ F))/ (P(E))  ("since"  E ∩ F = F ∩ E) `
Thus, P(E ∩F) = P(E). P(F|E)       .... (2) 
Combining (1) and (2), we find that 
P(E ∩ F) = P(E) P(F|E) 
= P(F) P(E|F) provided P(E) ≠ 0 and P(F) ≠ 0. 
The above result is known as the multiplication rule of probability. 
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