Also, we exclude the option that two bishops can inhabit the same square on the chessboard.
I've done this problem before with two rooks and it was fine, but I'm unsure how to sort it out with bishops. I wrote a short program to compute the probability by simulation and I got that the bishops will attack each other just less than 14% of the time. Is this correct? I'd like to understand this problem mathematically – could anyone please help?
Best Answer
I am assuming the bishops are indistinguishable, then the total number of ways you can put them is $\binom{8\times 8}{2},$ now attacking each other means that they are in the same diagonal, from $2$ to $7$ enumerate diagonals from any corner, then the total ways you can do this is $$2\left (2\sum _{n=2}^7\binom{n}{2}+\binom{8}{2}\right ),$$ where we multiply $2$ because there are $2$ possible orientations.The probability, which is the quotient, is $0.13889.$