Picking Among Two Pool Balls

[BMAD]

A cloth bag contains a pool ball, which is known to be a solid ball. A second pool ball is chosen at random in such a way that it is equally likely to be a solid or a stripe ball. The ball is added to the bag, the bag is shaken, and a ball is drawn at random. This ball proves to be a solid. What is the probability that the ball remaining in the bag is also a solid?

[bonanova]

Two-thirds.

Placing the unknown ball into the bag creates two situations with equal likelihood.
Let O denote a solid ball and I denote a stripe ball. Then your two possibilities are

1. O O
2. O I

There are three O's here, equally likely to have been drawn.

Two of them have another O in the same bag.