Hello, thereddevils!

After the exchange, bag A has been restored to its original state.Bag A contains 3 red balls and 2 green balls.

Bag B contains 2 red balls and 3 green balls.

Two balls are withdrawn from bag A to be put into bag B.

Then two balls are taken out from bag B and put in bag A.

What is the probability that bag A now contains 3 red balls and 2 green balls?

Whatever colors were moved to bag B were subsequently returned to bag A.

There are 3 cases to consider:

. .

. .

. .

[1] Two Reds are moved both ways.

. .

. . Bag B now contains: 4 Red, 3 Green

. .

. . Hence: .

[2] A Red and a Green are moved both ways.

. .

. . Bag B now contains: 3 Red, 4 Green.

. .

. . Hence: .

[3] Two Greens are moved both ways.

. .

. . Bag B now contains: 2R, 5G.

. .

. . Hence: .

Therefore: .