Question

On a chess-board if two squares are chosen, the probability that they have a side in common is

• A. $\frac{1}{9}$
• B. $\frac{1}{18}$
• C. $\frac{1}{27}$
• D. None of these.

Answer 1

The correct option is B

$\frac{1}{18}$

Explanation for the correct option:

Finding the probability that they have a side in common if two squares are chosen:

Calculate probability of selecting two squares.

There are total $64$ squares ,

From them the number of ways to select two squares is , ${}^{64}{C}_2=32\times 63.$

Calculate the possible ways and the needed probability.

Now, Each corner has two neighbor squares

There are $4$ corners

Total cases for this way $=4\times 2=8$

In borders of total $4$sides, $24$squares has $3$ neighbors each.

Total cases$=24\times 3=72.$

And for remaining $36$ squares has four neighbors in each.

Total cases $=36\times 4=144.$

So total cases will be $=8+72+144=224.$

Now probability of having a side in common is , $=\frac{224}{32\times 63}\times \frac{1}{2}=\frac{1}{18}$

Thus, the probability of having a side in common is $\frac{1}{18}$​.

Hence the correct answer is option (B)