If the mean and variance of a binomially distributed random variable…

The correct answer is Option (3) → \(\displaystyle\frac{7}{64}\)

For a binomial distribution B(n, p), mean = np and variance = npq, with q = 1 − p.

    \(np = 4,\quad npq = 2\)

Dividing:

    \( \displaystyle q = \frac{2}{4} = \frac{1}{2},\quad p = \frac{1}{2} \)

    \( \displaystyle n = \frac{4}{p} = 8 \)

Then

    \( \displaystyle P(X = 2) = {}^{8}C_{2}\left(\frac{1}{2}\right)^{2}\left(\frac{1}{2}\right)^{6} \)

    \( \displaystyle = \frac{28}{256} = \frac{7}{64} \)