Two smallest squares are chosen one by one on a chessboard. The probability that they have a side in common is:
Show Hint
- \(\frac{1}{9}\)
- \(\frac{2}{7}\)
- \(\frac{1}{18}\)
- \(\frac{5}{18}\)
The Correct Option is C
Solution and Explanation
Step 1: The problem asks for the probability that two randomly chosen smallest squares on a chessboard share a side. A standard chessboard consists of \(8 \times 8\) squares, which means there are 64 total small squares.
Step 2: The number of possible pairs of squares is given by \(\binom{64}{2}\).
Step 3: The number of favorable outcomes where two squares have a side in common is based on the number of adjacent squares. Each square has up to 4 adjacent squares (except for the edge squares), and the total number of such adjacent square pairs is fewer than 64, as we are restricted to squares that share a side.
Step 4: By counting the adjacent pairs (both horizontally and vertically) and dividing by the total number of pairs, we obtain the probability \(\frac{1}{18}\).
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