If \( A_1,A_2,A_3,\ldots A_8 \) are independent events such that \( P(A_i)=\displaystyle\frac{1}{1+i},1\le i\le 8 \), then the probability that none of the events occur, is
Answer & explanation
Correct answer: option 4
The correct answer is Option (4) → \( \displaystyle\frac{1}{9} \)
The events are independent, so their complements are too. Each
\( \displaystyle P(\overline{A_i})=1-\frac{1}{1+i}=\frac{i}{i+1} \)
\( \displaystyle P(\text{none})=\frac{1}{2}\times\frac{2}{3}\times\frac{3}{4}\times\cdots\times\frac{8}{9} \)
\( \displaystyle =\frac{1}{9} \)
(the product telescopes).
Why the other options are incorrect
Option (1) \( \displaystyle\frac{2}{9} \) — Incorrect. The product is \( \frac{1}{9} \), not \( \frac{2}{9} \).
Option (2) \( \displaystyle\frac{1}{3} \) — Incorrect. The product is \( \frac{1}{9} \), not \( \frac{1}{3} \).
Option (3) \( \displaystyle\frac{8}{9} \) — Incorrect. This is the probability that at least one event occurs.