Question

Two smallest squares are chosen one by one on a chessboard. The probability that they have a side in common is:

• A. \(\frac{1}{9}\)
• B. \(\frac{2}{7}\)
• C. \(\frac{1}{18}\)
• D. \(\frac{5}{18}\)

Hint:

To calculate probabilities in combinatorics, count the total number of possible outcomes and favorable outcomes, then divide the two.

Answer 1

The Correct Option is C

Step 1: We need to find the probability that two randomly selected squares on a chessboard share a side. A standard chessboard has \(8 \times 8 = 64\) squares.

Step 2: The total number of square pairs is \(\binom{64}{2}\).

Step 3: We determine favorable outcomes by counting adjacent square pairs. Each square has up to 4 neighbors. The total count of adjacent square pairs is less than 64, considering only squares that share a side.

Step 4: Calculating the ratio of adjacent pairs to total pairs yields the probability \(\frac{1}{18}\).