The probability that two randomly chosen squares on a standard 8×8 chessboard share a common side (i.e., are adjacent horizontally or vertically) is 118\frac{1}{18}181.
Explanation:
- Total ways to choose 2 squares: There are 64 squares on the chessboard. The total number of ways to choose any 2 squares is:
(642)=64×632=2016.\binom{64}{2}=\frac{64\times 63}{2}=2016.(264)=264×63=2016.
- Counting pairs of adjacent squares: Adjacent squares share a common side. We count these pairs by considering horizontal and vertical adjacencies:
- Horizontal adjacencies: Each of the 8 rows has 7 pairs of adjacent squares, so total horizontal pairs:
8×7=56.8\times 7=56.8×7=56.
* **Vertical adjacencies:** Each of the 8 columns has 7 pairs of adjacent squares, so total vertical pairs:
8×7=56.8\times 7=56.8×7=56.
Adding these gives the total number of adjacent pairs:
56+56=112.56+56=112.56+56=112.
- Probability calculation: Since the total number of pairs is 2016 and the number of adjacent pairs is 112, the probability that two randomly chosen squares share a common side is:
1122016=118.\frac{112}{2016}=\frac{1}{18}.2016112=181.
Alternative approach (by square types):
The chessboard squares can be classified into three types based on the number of adjacent squares:
- Corner squares (4 squares): Each has 2 adjacent squares.
- Edge squares excluding corners (24 squares): Each has 3 adjacent squares.
- Inner squares (36 squares): Each has 4 adjacent squares.
The probability can be computed by summing the probabilities for each type:
(3664×463)+(2464×363)+(464×263)=118.\left(\frac{36}{64}\times \frac{4}{63}\right)+\left(\frac{24}{64}\times \frac{3}{63}\right)+\left(\frac{4}{64}\times \frac{2}{63}\right)=\frac{1}{18}.(6436×634)+(6424×633)+(644×632)=181.
This matches the result above. Final answer:
118\boxed{\frac{1}{18}}181
This means there is about a 5.56% chance that two randomly chosen squares on a chessboard share a common side
