Given two bags A and B follows: bag A contains 3 red and 2 white balls and bag B contains 2 red and 5 white balls. A bag is selected at random, a ball is drawn and put into the other bag, and then a ball is drawn from the second bag. The probability that both balls drawn are of the same color is

The whole event consists of the following mutually exclusive ways.

(1) Selecting the bag A, drawing a red ball from A and putting it into bag B and then drawing a red ball from B.

(2) Selecting the bag A, drawing a white ball from A and putting it into bag B and then drawing a white ball from B.

(3) Selecting the bag B, drawing a red ball from B and putting it into A and then drawing a red ball from A.

(4) Selecting the bag B, drawing a white ball from B and putting it into A and then drawing a white ball from A.

The required probability is $=\frac{1}{2}\times \frac{3}{5}\times \frac{3}{8}+\frac{1}{2}\times \frac{2}{5}\times \frac{3}{4}+\frac{1}{2}\times \frac{2}{7}\times \frac{2}{3}+\frac{1}{2}\times \frac{5}{7}\times \frac{1}{2}$

                                                 $=\frac{9}{80}+\frac{3}{20}+\frac{2}{21}+\frac{5}{28}=\frac{901}{1680}$