A father and son take turns picking red and green balls from a bag. There are 2 red balls and 3 green balls. The first person to pick a red ball wins. There is no replacement. What is the probability that the father wins if he goes first?
I drew a binary tree to solve this. The father can only win the first round and the third round.
P(father wins first round) = $\frac25$
P(father wins third round) = $\frac35 * \frac24 * \frac23 = \frac15$
P(father wins first round) + P(father wins third round) = $\frac25+\frac15 =\frac35$
Is this correct?