Question

Assertion :Given $E_i(i=1,2,.....n)$ are n independent events such that $P\left({\overline{E}}_i\right)=\frac{i}{i+1},1\le i\le n$, the probability that none of the n events occur is $\frac{n}{n+1}$ Reason: Probability of independent events is equal to the product of the probabilities of these 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 the ,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

$P\left({\overline{E}}_i\right)=\frac{i}{1+i}(i=1,2,\mathrm{3...},n)$

$P\left({\overline{E}}_1\right)=\frac{1}{2},P\left({\overline{E}}_2\right)=\frac{2}{3},P\left({\overline{E}}_3\right)=\frac{3}{4},...P\left({\overline{E}}_n\right)=\frac{n}{n+1}$

$\therefore P({\overline{E}}_1{\overline{E}}_2{\overline{E}}_3....{\overline{E}}_n)=\frac{1}{2}.\frac{2}{3}.\frac{3}{4}...\frac{n-1}{n}\cdot \frac{n}{n+1}=\frac{1}{n+1}$
$\therefore$ Assertion (A) is false but Reason (R) is true.