check and mate !
Probability
Level
2
Two squares are chosen at random on a chessboard. What is the probability that they have a side in common ?
0.312
0.056
0.5
9.8E-3
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1 solution
Its clear that we have to choose 2 squares out of 64 squares having a side common. So if we choose first square the no of ways are 64; for other square no of ways becomes 63. Total no of ways= 64 × 63= 4032.
So either of the ways are possible :
1) we choose 4 corner squares each square shares a side with 2 squares. No of ways = 4 × 2. OR
2) we choose remaining squares only on the outer sides of the chess board excluding 4 corner squares. Each square shares a common side with with 3 other squares. So No of ways= 24 × 3. OR
3) we choose remaining 36 squares having a side common with 4 other squares . No of ways = 36 × 4
So the required probability according to the question becomes (4 × 2) + (24 × 3)+ (36 × 4) = 224. 224/4032 = 0.056.