Mean and variance of a binomial distribution are 6 and 2 respectively…

The correct answer is Option (2) → \(\displaystyle\frac{16}{3^7}\)

For a binomial distribution, mean = np and variance = npq. Dividing the variance by the mean:

    \( \displaystyle q = \frac{npq}{np} = \frac{2}{6} = \frac{1}{3} \)

    \( \displaystyle p = 1 - \frac{1}{3} = \frac{2}{3} \)

    \( \displaystyle n \cdot \frac{2}{3} = 6 \)

    \(n = 9\)

Then

    \( \displaystyle P(X = 2) = {}^{9}C_{2}\left(\frac{2}{3}\right)^2\left(\frac{1}{3}\right)^7 \)

    \( \displaystyle P(X = 2) = 36 \cdot \frac{4}{9} \cdot \frac{1}{3^7} = \frac{16}{3^7} \)