For independent events \(A_1, A_2, A_3, \ldots, A_n\), if \(P(A_i) = \displaystyle\frac{1}{i+1}, i = 1, 2, 3, \ldots, n\), then the probability that none of the events occur is:
Answer & explanation
Correct answer: option 3
The correct answer is Option (3) → \(\displaystyle\frac{1}{n+1}\)
“None of the events occurs” means every complement \(A_i'\) occurs. The complements of independent events are also independent, so their probabilities multiply:
\( \displaystyle P(A_i') = 1 - \frac{1}{i+1} = \frac{i}{i+1} \)
\( \displaystyle P(\text{none}) = \frac{1}{2} \times \frac{2}{3} \times \frac{3}{4} \times \cdots \times \frac{n}{n+1} \)
Each numerator cancels the denominator before it (a telescoping product), leaving
\( \displaystyle P(\text{none}) = \frac{1}{n+1} \)