If A and B are two events in a random experiment such that P(A) ≠ 0 and P(B) ≠ 0, then the probability of the simultaneous occurrence of the events A and B i.e., P(A ∩ B) is given by:
\(P(A\,\cap\,B)=P(A)\times P(B/A) \,or\,P(A\,\cap\,B) = P(B)\times P(A/B) \)
(This follows directly from the formula given for conditional probability in Key Fact No. 1)
Thus, the above-given formulae hold true for dependent events.
Corollary 1: In the case of independent events, the occurrence of event B does not depend on the occurrence of A, hence P(B/A) = P(B).
∴ \(P(A\,\cap\,B) = P(A)\times P(B)\)
Thus, we can say if \(P(A\,\cap\,B) = P(A)\times P(B)\), then the events A and B are independent.
Also, If A and B are two independent events associated with a random experiment having a sample space S, then
(a) \(\overline{A}\) and B are also independent events. So,
\(P(\overline{A}\,\cap\,B)= P(\overline{A})\times P(B)\)
(b) A and \(\overline{B}\) are also independent events, so,
\(P(A\,\cap\,\overline{B}) =P(A)\times P(\overline{B})\)
(c) \(\overline{A}\)and \(\overline{B}\) are also independent events, so,
\(P(\overline{A}\,\cap\,\overline{B})= P(\overline{A})\times P(\overline{B})\)
Corollary 2: If A1, A2, A3, ..., An are n independent events associated with a random experiment, then
\(P(A_1\,\cap A_2 \cap A_3,...,\cap A_n)= P(A_1)\times P(A_2)\times P(A_3) ...\times P(A_n)\)
Corollary 3: If A1, A2, A3, ..., An are n independent events associated with a random experiment, then
\(P(A_1\cup A_2 \cup A_3\dotsb \cup A_n) = 1-P(\overline{A_1})\times P(\overline{A_2})\times P(\overline{A_3})\times ... \times P(\overline{A_n})\)
Corollary 4: If the probability that an event will happen is p, the chance that it will happen in any succession of r trials is pr .
Also for the r repeated non-occurrence of the event we have the probability = (1 – p)r.
