Question

Assertion: Given $E_i(i=1,2,.......n)$ are $n$ independent events such that $P\left(\overline{E_i}\right)=\frac{i}{i+1},1\le i\le n$, then the probability that none of the $n$ events occur is $\frac{n}{n+1}$

Reason: Probability of occurence of all independent events is equal to the product of the probabilities of individual events.

• A. Both Assertion & Reason are individually true & Reason is correct explanation of Assertion
• B. Both Assertion & Reason are individually true but Reason is not correct (proper) explanation of Assertion
• C. Assertion is true but Reason is false
• D. Assertion is false but Reason is true

Answer 1

The correct option is D Assertion is false but Reason is true

$P\left(\overline{E_i}\right)=\frac{i}{1+i}(i=1,2,\mathrm{3....},n)$
$P\left(\overline{E_1}\right)=\frac{1}{2},P\left(\overline{E_2}\right)=\frac{2}{3},P\left(\overline{E_3}\right)=\frac{3}{4},....P\left(\overline{E_n}\right)=\frac{n}{n+1}$
$\therefore P(\overline{E_1}\cap \overline{E_2}\cap \overline{E_3}.....\cap \overline{E_n})=\frac{1}{2}.\frac{2}{3}.\frac{3}{4}....\frac{n-1}{n}.\frac{n}{n+1}=\frac{1}{n+1}$
$\therefore$ Assertion $\left(A\right)$ is false but Reason $\left(R\right)$ is true.