Find the probability that A wins after second draw

Question

A bag contains 5 white balls and 3 red balls. Two players A and B take turns at drawing one ball from the ball at random, and balls are not replaced. The player who first gets two red balls is the winner, and the drawing stops as soon as either player has drawn two red balls. Player A draws first. Find the probability that player A is the winner, given that the winning player wins on his second draw.

Answer 1

The play goes:

A draws first.

B draws second.

A draws again and wins. But A can only win with two red balls, so A drew a red ball on the first draw. B could not have drawn a red on his first draw.

Now the probabilities. With 3 out of 8 balls being red, the first draw probability of red=3/8.

B doesn't draw a red ball so the probability of this is 5/7 because there are only 7 balls left and 5 are white.

A draws a red ball on his second turn. Only 6 balls remain and there are 4 white balls and 2 red. So the probability is 2/6=1/3.

The combined probabilities are 3/8*5/7*1/3=5/56.