For independent events $A_1, A_2, A_3, ..., A_n$, if $P(A_i)=\frac{1}{i+1},i= 1,2,3,..., n$, then the probability that none of the events occur is:
Answer & explanation
Correct answer: option 3
The correct answer is Option (3) → $\frac{1}{n+1}$
Given: Independent events $A_1, A_2, \dots, A_n$ with $P(A_i) = \frac{1}{i+1}$
Probability that none of the events occur:
$P(\text{none}) = P(A_1') P(A_2') \cdots P(A_n')$
$P(A_i') = 1 - P(A_i) = 1 - \frac{1}{i+1} = \frac{i}{i+1}$
Therefore,
$P(\text{none}) = \frac{1}{2} \cdot \frac{2}{3} \cdot \frac{3}{4} \cdots \frac{n}{n+1}$
Telescoping product:
$P(\text{none}) = \frac{1 \cdot 2 \cdot 3 \cdots n}{2 \cdot 3 \cdot 4 \cdots (n+1)} = \frac{1}{n+1}$
Required probability = $\frac{1}{n+1}$