Multiplication Theorem of Probability

This theorem states that if two events \(A\) and \(B\) are independent, the probability that they both will occur is equal to the product of their individual probability. Symbolically, if \(A\) and \(B\) are independent, then \[P(A\ \text{and}\ B)=P\ (A) \times\ P\ (B)\] The theorem can be extended to three or more independent events. Thus, \[P(A,\ B\ \text{and}\ C)=P\ (A)\ \times\ P\ (B)\ \times\ P\ (C)\] Proof of theorem

If an event \(A\) can happen in \(n_1\) ways of which \(a_1\) are successful and the vent \(B\) can happen in \(n_2\) ways of which \(a_2\) are successful. We can combine each successful event in the first with each successful event in the second case. Thus, the total number of successful happenings in both cases is \(a_1 \times a_2\). Similarly, the total number of possible cases is \(n_1 \times n_2\).

Then by definition the probability of the occurrence of both events is \[\frac{a_1 \times a_2}{n_1 \times n_2}= \frac{a_1}{n_1} \times \frac{a_2}{n_2}\] But, \[\frac{a_1}{n_1}=P(A)\] and \[\frac{a_2}{n_2}=P(B)\] \(\therefore\) \[P(A\ \text{and}\ B)=P(A) \times P(B)\] Similarly it can be extended to three or more events.