(a) The probability of obtaining a pair of heads (HH) in one throw is \(\frac{1}{4}\). The expected number of throws \(E(X)\) is the reciprocal of this probability, so \(E(X) = \frac{1}{\frac{1}{4}} = 4\).

(b) The probability that exactly 5 throws are required is given by the geometric distribution formula: \(P(X = 5) = \left( \frac{3}{4} \right)^4 \left( \frac{1}{4} \right) = 0.0791\).

(c) The probability that fewer than 7 throws are required is \(P(X < 7) = 1 - \left( \frac{3}{4} \right)^6\). Calculating this gives \(1 - \left( \frac{3}{4} \right)^6 = 0.822\).