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Random Chess Pieces Poser (Posted on 2016-08-29) Difficulty: 3 of 5
A queen, a rook and a bishop are randomly placed on distinct squares of an ordinary chessboard. Find the probability that:.
(i) The queen is under attack from either the bishop or the rook.
(ii) The bishop is neither under attack from the queen, nor under attack from the rook.

No Solution Yet Submitted by K Sengupta    
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re(2): computer aided solution | Comment 4 of 6 |
(In reply to re: computer aided solution by Jer)

I'm with Jer on this.  I decided not to tackle it when I noticed the blocking issue.


On Part I,

The probability that the Q is in the same diagonal as the bishop is 10/63.

The probability that the Q is on the same rank or file as the rook is 14/63.

If they could attack through each other, than the answer would be 10/63 + 14/63 = 24/63 = 8/21 ~~ 38%.

So Charlie's answer of ~~ 30%, considering blocking, is not crazy.  Personally, though, it seems too low.  If I am right, it implies that 20% of the time that the rook is on the same rank or file, the bishop is between them, and vice versa, and this can't be right.

So I think that either Charlie or myself or both is wrong.

Actually, I am pretty sure that my 38% is a little high, as the bishop and the rook are not independent.  If the rook is not on the same rank and file, then it decreases slightly the probability that the bishop is on the same diagonal, and vice versa.

But I still suspect Charlie's answer is too low.  

  Posted by Steve Herman on 2016-08-30 09:00:30
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