Chess board problem!

First , find the no. of ways in which the two queens can attack each other :

• When two queens are in same horizontal line : $\dbinom{8}{2}\times 8$

• When two queens are in same vertical line : $\dbinom{8}{2}\times 8$

• When two queens are diagonal to each other : $2\times \left( \dbinom{9}{3}+\dbinom{8}{3}\right)$

Total ways = $588$

Hence the required probability is :

$1-\dfrac{588}{\dbinom{64}{2}}$

$=\boxed{\dfrac{23}{36}}$