(a) To find \(P(X = 3)\), we use the binomial probability formula:

\(P(X = 3) = \binom{4}{3} \left( \frac{1}{4} \right)^3 \left( \frac{3}{4} \right)^1\)

\(= 4 \times \frac{1}{64} \times \frac{3}{4}\)

\(= \frac{3}{64}\)

(b) To complete the probability distribution table, we calculate:

\(P(X = 1) = \binom{4}{1} \left( \frac{1}{4} \right)^1 \left( \frac{3}{4} \right)^3 = \frac{27}{64}\)

\(P(X = 2) = \binom{4}{2} \left( \frac{1}{4} \right)^2 \left( \frac{3}{4} \right)^2 = \frac{27}{128}\)

The completed table is:

(c) To find \(E(X)\), we calculate the expected value:

\(E(X) = 0 \times \frac{81}{256} + 1 \times \frac{27}{64} + 2 \times \frac{27}{128} + 3 \times \frac{3}{64} + 4 \times \frac{1}{256}\)

\(= 0 + \frac{27}{64} + \frac{54}{128} + \frac{36}{256} + \frac{4}{256}\)

\(= 1\)