#### 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.

Video link : https://youtu.be/DrfBeX1t_r4