Eric has three coins. One of the coins is fair. The other two coins are each biased so that the probability of obtaining a head on any throw is \(\frac{1}{4}\), independently of all other throws. Eric throws all three coins at the same time.

Events \(A\) and \(B\) are defined as follows.

\(A\): all three coins show the same result

\(B\): at least one of the biased coins shows a head

(a) Show that \(P(B) = \frac{7}{16}\).

(b) Find \(P(A \mid B)\).

The random variable \(X\) is the number of heads obtained when Eric throws the three coins.

(c) Draw up the probability distribution table for \(X\).