Box $A$ contains $3$ red and $2$ black balls. Box $B$ contains $2$ red and $3$ black balls. One ball is drawn at random from box$ A$ and placed in box $B$. Then one ball is drawn at random from the box $B$ and placed in $A$. The probability that the composition of balls in the two boxes remains unaltered is

The composition remains unaltered if a red ball is transferred from A to B and a red ball is returned from B to A, or if a black ball is transferred from A to B and a black ball is returned from B to A. The first probability is (3/5) * (3/6) = 9/30, and the second is (2/5) * (4/6) = 8/30. Adding these mutually exclusive probabilities gives 9/30 + 8/30 = 17/30.

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