Question

For a binomial variate \(X\) if \(n=5\) and \[ P(X=1)=8P(X=3), \] then its variance is

• A. \(\dfrac{1}{5}\)
• B. \(\dfrac{4}{5}\)
• C. \(\dfrac{3}{5}\)
• D. \(\dfrac{2}{5}\)

Hint:

For a binomial distribution, \[ \boxed{ \text{Variance}=npq, } \] where \[ \boxed{q=1-p.} \]

Answer 1

The Correct Option is B

Step 1: Write the binomial probabilities. Let the probability of success be \(p\), so that \(q=1-p\). \[ P(X=1)=\binom51pq^4=5pq^4, \] \[ P(X=3)=\binom53p^3q^2=10p^3q^2. \] Given, \[ 5pq^4 = 8(10p^3q^2). \]

Step 2: Find \(p\). Simplifying, \[ q^2=16p^2 \] or \[ q=4p. \] Since \[ p+q=1, \] \[ p+4p=1 \] \[ p=\frac15,\qquad q=\frac45. \] The variance of a binomial distribution is \[ npq = 5\left(\frac15\right)\left(\frac45\right) = \frac45. \] Hence, \[ \boxed{\frac45} \] Therefore, \[ \boxed{(B)} \] is the correct answer.