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 A_{1}, A_{2}, A_{3}, ..., A_{n} 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 A_{1}, A_{2}, A_{3}, ..., A_{n }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 p^{r} .

Also for the** r repeated non-occurrence** of the event we have the probability = (1 – p)^{r}.