(a) To find the probability of obtaining exactly one head, consider two scenarios: the biased coin shows a head and the three fair coins show tails, or the biased coin shows a tail and one of the fair coins shows a head.

Scenario 1: Biased coin shows head, three fair coins show tails:

\(Probability = 0.6 × (0.5)^3 = 0.6 × 0.125 = 0.075\)

Scenario 2: Biased coin shows tail, one fair coin shows head:

\(Probability = 0.4 × (0.5)^3 × 3 = 0.4 × 0.125 × 3 = 0.15\)

\(Total probability = 0.075 + 0.15 = 0.225\)

(b) To complete the probability distribution table, use the given probabilities and the fact that the sum of probabilities must equal 1.

\((c) Given E(X) = 2.1, use the variance formula:\)

\(Var(X) = E(X^2) - [E(X)]^2\)

Calculate E(X^2) using the probabilities:

\(E(X^2) = 0^2 × 0.05 + 1^2 × 0.225 + 2^2 × 0.375 + 3^2 × 0.275 + 4^2 × 0.075\)

\(= 0 + 0.225 + 1.5 + 2.475 + 1.2 = 5.4\)

\(Var(X) = 5.4 - (2.1)^2 = 5.4 - 4.41 = 0.99\)