Question

If $X_1, X_2, ..., X_n$ are $n$ independent events such that $P(X_r) = \frac{1}{r+1}$, for $r=1, 2, ..., n$, then the probability that none of the $n$ events occur is

• A. $\frac{1}{n}$
• B. $\frac{1}{n+1}$
• C. $\frac{n}{n+1}$
• D. $\frac{n+1}{n+2}$

Hint:

Telescoping Products in Probability. Products like $\prod_r=1^n \fracrr+1$ simplify beautifully due to cancellation.

Answer 1

The Correct Option is B

We are given $n$ independent events $X_1, X_2, \dots, X_n$ with $P(X_r) = \frac{1}{r+1}$. \[ P(\text{none occur}) = \prod_{r=1}^{n} (1 - P(X_r)) = \prod_{r=1}^{n} \left(1 - \frac{1}{r+1}\right) = \prod_{r=1}^{n} \frac{r}{r+1} \] Telescoping product: \[ \frac{1}{2} \cdot \frac{2}{3} \cdot \frac{3}{4} \cdots \frac{n}{n+1} = \frac{1}{n+1} \]