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If a rook and a bishop of a standard chess set are randomly placed on a chessboard...(read details)?
If a rook and a bishop of a standard chess set are randomly placed on a chessboard, what is the probability that one is attacking the other? Note that both are different colored pieces..
Regardless of where the rook is, there are exactly fourteen squares the rook is attacking (the seven squares in the same rank, and the seven squares in the same file)
A bishop attacks:
* Seven squares if it is on one of the 28 squares on the border
* Nine squares if it is on one of the 20 squares that form a rectangle just inside the border squares
* Eleven squares if it is on one of the 12 squares that form a rectangle just inside the previous rectangle
* Thirteen squares if it is on one of the 4 center squares.
(It may be helpful to refer to a chessboard to verify that this is true. Alternatively, look at an image of a chessboard: http://www.cs.tau.ac.il/~turkel/chessboard.jpg )
Note that a rook and a bishop cannot attack each other simultaneously (because if the rook is attacking the bishop, then the two are either in the same rank or the same file, and thus are not diagonal from each other).
---
Thus, for two randomly places pieces, the probability that the rook is attacking the bishop is 14/63 (14 attacked squares out of 63 other possible squares the bishop can be on). This simplifies to 2/9.
The probability that the bishop is attacking the rook is
(28/64) * (7/63) + (20/64) * (9/63) + (12/64) * (11/63) + (4/64) * (13/63).
(This is because there are 28 squares out of 64 where the bishop attacks 7 squares, and for each of these there are 7 squares out of 63 that the rook could be on so the bishop attacks it; and similar reasoning holds for the other three terms in the sum.)
This expression simplifies to 5/36.
Since the two events are mutually exclusive (a rook and a bishop cannot attack each other simultaneously), then the probability that either one is attacking the other is the sum of the two probabilities above:
2/9 + 5/36 = 13/36.
This is about a 36.1% chance.
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