If A1- A2- -An are n independent events such that PAi-1-i-1- i-1- 2- -n- The probability that none of A1- A2- -An occurs is

P(A'∩A'_2.......∩A'_n) =P(A'_1)P(A'_2).....P(A'_n) [∵ A_1, A_2,....,A_n are independent] = ( 1 1/2 ) ( 1 1/3 )... ( 1 1/n+1 ) =1/2×2/3×3/4×...×n 1/n×n/n+1=1/n ...

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Standard XIII

Mathematics

Independent Events

If A1, A2, .....

Question

If $A_{1}, A_{2}, . . . . ., A_{n}$ are n independent events such that $P (A_{i}) = \frac{1}{i + 1}, i = 1, 2, . . . ., n$ . The probability that none of $A_{1}, A_{2}, . . . . A_{n}$ occurs is

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

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

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more than $\frac{1}{n}$

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None of these

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Solution

The correct option is B
$\frac{1}{n + 1}$

$P (A^{'} \cap A_{2}^{'} . . . . . . . \cap A_{n}^{'}) = P (A_{1}^{'}) P (A_{2}^{'}) . . . . . P (A_{n}^{'})$
$[∵ A_{1}, A_{2}, . . . ., A_{n} are independent]$
$= (1 - \frac{1}{2}) (1 - \frac{1}{3}) . . . (1 - \frac{1}{n + 1}) = \frac{1}{2} \times \frac{2}{3} \times \frac{3}{4} \times . . . \times \frac{n - 1}{n} \times \frac{n}{n + 1} = \frac{1}{n + 1} < \frac{1}{n}$

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If A1,A2,.....,An are n independent events such that P(Ai)=1i+1,i=1,2,....,n. The probability that none of A1,A2,....An occurs is

The correct option is B 1n+1 P(A′∩A′2.......∩A′n)=P(A′1)P(A′2).....P(A′n) [∵A1,A2,....,An are independent] =(1−12)(1−13)...(1−1n+1)=12×23×34×...×n−1n×nn+1=1n+1<1n

If $A_{1}, A_{2}, . . . . ., A_{n}$ are n independent events such that $P (A_{i}) = \frac{1}{i + 1}, i = 1, 2, . . . ., n$ . The probability that none of $A_{1}, A_{2}, . . . . A_{n}$ occurs is