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:

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

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