A castle (usually known as a rook) moves laterally and a bishop moves diagonally. That means that they can never both threaten each other simultaneously. So the probability that one piece threatens the other is equal to the probability that the rook threatens the bishop, plus the probability that the bishop threatens the rook.
A rook placed anywhere on the board threatens 14 squares. The probability that the rook threatens the bishop is 14/63, or 2/9.
The bishop . . .
Threatens 7 squares when placed on one of the 28 edge squares.
Threatens 9 squares when placed on one of the 20 squares one square from the edge.
Threatens 11 squares when placed on one of the 12 squares two squares from the edge.
Threatens 13 squares when placed on one of the 4 center squares.
The average number of squares threatened by a bishop is
$$\dfrac{\left(7\times28 \right ) + \left(9\times20 \right ) + \left(11\times12 \right ) + \left (13\times4 \right )}{64}\; = \; \dfrac{35}4 \\\\
\small \text {The probability that the bishop threatens the rook} \;\dfrac{35}{4\times63} = \dfrac5{36}\\\\
\text {The probability that either threatens the other } \;
\dfrac{5}{36}+ \dfrac{2}{9} \;=\; \dfrac{13}{36}\\$$
Here you can see that getting hooked by a rook is 62.5% more likely than getting buggered by a bishop.
I’m not sure where the knight came in. Maybe Cphill is taking a break from his quest.
