I'm sure there must be a problem like this posted already, apologize for the duplicate- all the names are so similar I can't find one sifting through. The problem goes like this:
Urn 1 contains 3 white balls and one red ball. Urn 2 contains 2 white balls and 3 red balls.
a) An urn is chosen at random, and a ball is drawn from it at random. If the ball drawn is R, what is the probability the urn now contains no red balls?
answer: .25, which I'm pretty confident is right. Feel free to correct me.
This is the part where I'm stuck.
b) An urn is is chosen at random which will give a ball, and the other one will receive that ball. Then, a ball is drawn at random from the urn the received a ball. Given that this ball drawn is red, what is the probability that the donating urn now contains no red balls?
It seems like this should also just be $\frac13$, since there are only 3 balls that can be drawn initially resulting in in an urn with 0 reds; the 2 whites drawn from urn 2, or the red drawn from urn 1. Is this correct?