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Probability question

2021 · 1 Sep · Shift 2 · Q27
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  5. /2021 · 1 Sep · Shift 2 · Q27

Probability question

2021 · 1 Sep · Shift 2 · Q27

JEE MainMathematicsProbabilityMCQ+4 / −1
Two squares are chosen at random on a chessboard (see figure). The probability that they have a side in common is : JEE Main 2021 (Online) 1st September Evening Shift Mathematics - Probability Question 108 English
  1. A
    27{2 \over 7}72​
  2. B
    118{1 \over 18}181​
  3. C
    17{1 \over 7}71​
  4. D
    19{1 \over 9}91​
View written solutionFree

Correct answer: B

  1. Total number of squares on a chessboard

A standard chessboard has 8×8=648 \times 8 = 648×8=64 small squares.

Two squares are chosen at random, so the total number of ways is (642)=64⋅632=2016.\binom{64}{2} = \frac{64\cdot 63}{2} = 2016.(264​)=264⋅63​=2016.

  1. Count pairs of squares having a common side

Two squares have a side in common if they are adjacent either:

  • horizontally, or
  • vertically.

Horizontal adjacent pairs

In each row of 8 squares, there are 7 horizontal adjacent pairs. Since there are 8 rows, 8⋅7=568 \cdot 7 = 568⋅7=56 horizontal pairs.

Vertical adjacent pairs

Similarly, in each column of 8 squares, there are 7 vertical adjacent pairs. Since there are 8 columns, 8⋅7=568 \cdot 7 = 568⋅7=56 vertical pairs.

So total favorable pairs: 56+56=112.56+56=112.56+56=112.

  1. Compute the probability

Therefore, P=1122016.P = \frac{112}{2016}.P=2016112​. Simplify: 1122016=118.\frac{112}{2016} = \frac{1}{18}.2016112​=181​.

  1. Match with options

Thus the required probability is 118.\boxed{\frac{1}{18}}.181​​. So the correct option is B.

  1. Comparison with stored answer

Stored correct answer: B

Our derived answer is also B, so they agree.

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