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What is the Multiplication Theorem of Probability?

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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.

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