Two statements are given, one labelled Assertion (A) and the other labelled Reason (R).

Assertion (A): If the difference between mean and variance of a binomial distribution is 1 and the difference of their squares is 5, then the probability of success is $\frac{1}{3}$.

Reason (R): For a binomial distribution of n trials, mean = np and variance = npq, where p = probability of success and q = probability of failure.

Select the correct answer from the options given below.

Both Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of the Assertion (A).

The correct answer is Option (1) → Both Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of the Assertion (A).

We know that in a Binomial distribution, mean = $np$ and variance = $npq$.

Given $np - npq = 1$   ...(i)

and $(np)^2- (npq)^2 = 5$

$⇒(np - npq) (np + npq) = 5$

$⇒1 × (np + npq) = 5$   ...(using (i))

$⇒np + npq = 5$   ...(ii)

Solving (i) and (ii), we get

$np = 3$ and $npq = 2$.

So, $\frac{npq}{np}=\frac{2}{3}⇒q=\frac{2}{3}$

$p=1-q=p=1-\frac{2}{3}=\frac{1}{3}$

∴ Assertion is true.

Also, Reason is true and

Reason is the correct explanation of Assertion.