A chessboard has 8 rows and 8 columns. A rook and a bishop are put randomly and with equal probability among all pairs of squares of the board.
a. What is the probability that the two pieces are put on the squares (1,1) and (1,2)?
b. What is the probability that the two pieces are put on the same row?
8*8 = 64 squares
first piece has two possible spots
so
2/64 is chance it will pick one of them
now there are 63 spots left and the second piece has to hit it.
1/63 is that probability
2/64 * 1/63 = 4.96*10^-4 = .000496
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8 rows
first piece can hit any row but leaves that row with 7 empty spaces
second piece 7/63 chance that it hits that row
so 7/63
To find the probability of the rook and bishop being placed on specific squares or in the same row on the chessboard, we need to determine the total number of possible outcomes and the favorable outcomes.
a. Probability of rook on square (1,1) and bishop on square (1,2):
To calculate the probability, we need to first determine the total number of possible positions for the rook and bishop on the chessboard. Since each piece can be placed on any of the 64 squares, the total number of possibilities is 64 * 64 = 4096.
Now, the favorable outcome is when the rook is placed on square (1,1) and the bishop is placed on square (1,2). There is only one favorable outcome.
Therefore, the probability is: 1/4096 ≈ 0.00024
b. Probability of the two pieces being on the same row:
To calculate this probability, we need to determine the total number of possible positions for the rook and bishop on the chessboard and the favorable outcomes where they are on the same row.
The rook can be placed on any of the 64 squares, and the bishop can be placed on any of the remaining 63 squares. However, since the rook and bishop need to be on the same row, there are only 8 possible rows where they can be placed.
Therefore, the total number of possible outcomes is 64 * 63 = 4032, and the favorable outcomes, where the rook and bishop are on the same row, is 8 * 64 = 512.
Thus, the probability is: 512/4032 ≈ 0.127.