For the first part, you just calculate in order the odds of what HAS to happen for player A to win on his second draw.

So, first he has to draw a red ball, the odds of which are 3/8. Then, in one situation Player B draws a white, which is a 5/7 chance. Lastly A has to choose another Red, which would be a 2/6 chance.

However, B could have chosen a red ball on his draw, the odds of which were 2/7. Then, for B to get the last red ball would have been a 1/6 chance. So the total chance is