Step-by-step Solution:

Concept:

The binomial distribution of a random variable X is given by,

P(X = k) = \( \rm ^nC_k p^k q^{n-k} \)

Mean = np 

Variance = npq

Calculations:

Given, the mean and variance of a random variable X having a binomial distribution are 4 and 2 respectively,

⇒ mean = np = 4         ....(1)

⇒ variance = npq = 2    ....(2)

From equation (1) and (2), we have

⇒ q = \( \rm \dfrac 12 \)

We know, p = 1 - q 

⇒ p = \( \rm \dfrac 12 \)

From equation (1), we have

n = 8 

The binomial distribution of a random variable X is given by,

P(X = k) = \( \rm ^nC_k p^k q^{n-k} \)

⇒ P(X = 1) = \( \rm ^8C_1 p^1 q^{8-1} \)

⇒ P(X = 1) = 8. \( \rm \dfrac 12 \). \( \rm \dfrac {1}{2^7} \)

⇒ P(X = 1) = \( \rm \dfrac {1}{32} \)