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.
Answer & explanation
Correct answer: option 1
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.