If X_1, X_2, ldots, X_n are n independent eve | vedclass

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(B) The probability that none of the $n$ independent events $X_1, X_2, \ldots, X_n$ occur is given by $P(X_1' \cap X_2' \cap \ldots \cap X_n')$.Since

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$\frac{1}{n+1}$

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(B) The probability that none of the $n$ independent events $X_1, X_2, \ldots, X_n$ occur is given by $P(X_1' \cap X_2' \cap \ldots \cap X_n')$.Since the events are independent,this is equal to $P(X_1') P(X_2') \ldots P(X_n')$.Given $P(X_r) = \frac{1}{r+1}$,the probability of the complement event is $P(X_r') = 1 - P(X_r) = 1 - \frac{1}{r+1} = \frac{r}{r+1}$.Thus,the required probability is:$P(X_1') P(X_2') \ldots P(X_n') = \left(1 - \frac{1}{2}ight) \left(1 - \frac{1}{3}ight) \ldots \left(1 - \frac{1}{n+1}ight)$$= \frac{1}{2} 	imes \frac{2}{3} 	imes \frac{3}{4} 	imes \ldots 	imes \frac{n}{n+1}$$= \frac{1}{n+1}$.

If $X_1, X_2, \ldots, X_n$ are $n$ independent events such that $P(X_r) = \frac{1}{r+1}$ for $r = 1, 2, \ldots, n$,then the probability that none of the $n$ events occur is

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