I apologize for any confusion. Let's go through each probability calculation step-by-step:
a) Probability that the two pieces lie on the same row:
- There are a total of 64 unoccupied squares on the chessboard for the second piece to be placed.
- Since the first piece is already placed on a random square, there are 8 possible squares in the same row for the second piece to be placed.
- Therefore, the probability that the two pieces lie on the same row is 8/64, which simplifies to 1/8.
b) Probability that the pieces lie on the same row or column:
- To calculate this probability, we need to consider the number of squares in the same row and column that the second piece can be placed on.
- Since there are 8 rows and 8 columns on the chessboard, there are a total of 16 possible squares (8 in the same row and 8 in the same column) for the second piece to be placed.
- However, we need to subtract 1 square from this count because the square where the first piece is placed is already occupied.
- Therefore, the probability that the pieces lie on the same row or column is (16 - 1)/64, which simplifies to 15/64.
c) Probability that the pieces are adjacent to each other:
- In this case, adjacent means horizontally, vertically, or diagonally neighboring squares.
- To calculate this probability, we need to consider the number of squares adjacent to the first piece.
- If the first piece is placed on one of the 4 corner squares, there are only 3 squares adjacent to it.
- If the first piece is placed on one of the 4 edge squares (excluding the corners), there are 5 squares adjacent to it.
- If the first piece is placed on any of the remaining 36 squares in the center of the board, there are 8 squares adjacent to it.
- Therefore, the total number of favorable outcomes is (4 corners * 3 adjacent squares) + (4 edge squares * 5 adjacent squares) + (36 center squares * 8 adjacent squares) = 12 + 20 + 288 = 320.
- The total number of possible outcomes remains the same, which is 64 (the number of unoccupied squares on the board).
- Hence, the probability that the pieces are adjacent to each other is 320/64, which simplifies to 5/8.
In summary:
a) The probability that the two pieces lie on the same row is 1/8.
b) The probability that the pieces lie on the same row or column is 15/64.
c) The probability that the pieces are adjacent to each other is 5/8.
If you have a different set of correct answers, please provide the necessary information so I can assist you further.