Wherever the rook is placed on the board, it threatens 14 squares (7 on its rank and 7 on its file). So if the rook and the bishop are placed on different squares, the probability that the rook threatens the bishop is 14/63, or 2/9.
For the bishop, the situation is more complicated. If it is placed on one of the 28 squares on the edge of the board, it threatens 7 squares. If it is on one of the 20 squares one rank or file away from the edge, it threatens 9 squares. On one of the 12 squares that are two ranks or files from the edge, it threatens 11 squares. Finally, if it is on one of the 4 central squares, it threatens 13 squares. So the average number of squares threatened by a bishop is . Therefore the probability that the bishop threatens the rook is .
Adding those results together, you see that the probability that one piece threatens the other is .