ISI2017-DCG-22

Question

Let $A_1,A_2,A_3, \dots , A_n$ be $n$ independent events such that $P(A_i) = \frac{1}{i+1}$ for $i=1,2,3, \dots , n$. The probability that none of $A_1, A_2, A_3, \dots , A_n$ occurs is

• A. $\frac{n}{n+1}$
• B. $\frac{1}{n+1}$
• C. $\frac{n-1}{n+1}$
• D. none of these

Answer 1

Given $P(A{_{i}})= \frac{1}{i + 1}$

Since all events are independent, probability that none of them occurs is

$(1-P(A{_{1}})).(1-P(A{_{2}})).(1-P(A{_{3}}))\ldots(1-P(A{_{n}}))$

$\qquad =(1-\frac{1}{2}).(1-\frac{1}{3}).(1-\frac{1}{4})\ldots (1-\frac{1}{n+1})$

$\qquad =(\frac{1}{2}).(\frac{2}{3}).(\frac{3}{4})\ldots (\frac{n-1}{n}).(\frac{n}{n+1})$

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

Option (B) is correct

Comments

Isn't it 1/n+1 as n-1 also gets canceled. So option B

OR n!/(n+1)! Which is 1/n+1.

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Yes I'm getting 1/n+1.

which is correct ?