# Multiplication Theorem on Probability

## Notes

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.

If you would like to contribute notes or other learning material, please submit them using the button below.

### Shaalaa.com

Probability part 12 (Multiplication theorem on Probability) [00:13:38]
S
0%