Question

If two smallest squares are chosen at random on a chess board then the probability of getting these squares such that they do not have a side in common is

• A. \( \frac{1}{18} \)
• B. \( \frac{5}{36} \)
• C. \( \frac{17}{18} \)
• D. \( \frac{7}{36} \)

Hint:

When a probability question asks for "at least one" or "not", it's often much simpler to calculate the probability of the complementary event (the event you *don't* want) and subtract it from 1.

Answer 1

The Correct Option is C

First, find the total number of ways to choose 2 squares from a chessboard.
A chessboard has 64 squares. The total number of ways is \( ^{64}C_2 \).
\( ^{64}C_2 = \frac{64 \times 63}{2} = 32 \times 63 = 2016 \).
It is easier to calculate the probability of the complementary event: the two squares *do* have a side in common. Then we subtract this from 1.
Let's find the number of pairs of squares with a common side (adjacent squares).
Adjacent pairs can be horizontal or vertical.
Number of horizontal adjacent pairs: In each of the 8 rows, there are 7 adjacent pairs. So, \( 8 \times 7 = 56 \) pairs.
Number of vertical adjacent pairs: In each of the 8 columns, there are 7 adjacent pairs. So, \( 8 \times 7 = 56 \) pairs.
Total number of adjacent pairs = \( 56 + 56 = 112 \).
The probability of choosing an adjacent pair is \( P(\text{adjacent}) = \frac{\text{Number of adjacent pairs}}{\text{Total pairs}} = \frac{112}{2016} \).
Let's simplify this fraction. Both are divisible by 112. \( 2016 / 112 = 18 \).
So, \( P(\text{adjacent}) = \frac{1}{18} \).
The probability that the squares do not have a side in common is the complement.
\( P(\text{not adjacent}) = 1 - P(\text{adjacent}) = 1 - \frac{1}{18} = \frac{17}{18} \).