# Probability that someone will pick a red ball first?

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?

• Yes, this is correct. Nov 27 '14 at 18:56
• By the way, you can get a multiplication dot, as in $\frac35\cdot\frac 23$, by typing \cdot.
– MJD
Nov 27 '14 at 19:21

Your proposed solution is exactly correct. Nice work.

To check your answer, you can use the same method to calculate the second player's probability of winning; it ought to be $1-\frac35 =\frac 25$. Let $P_i$ be the probability that the game ends in round $i$; you have calculated $P_1 + P_3 = \frac 35$. Then \begin{align} P_2 & = \frac35\cdot \frac 24 & = \frac 3{10}\\ P_4 & = \frac 35\cdot \frac24\cdot\frac 13\cdot \frac22 &= \frac1{10} \end{align}

So $P_2 + P_4 = \frac25$ as we expected.

Correct

An alternative approach is, out of all $\binom{5}{2}$ ways to place red (and green) balls in a line, count ways that place the second red ball when the first red ball is either the first or third ball in line.

$$\dfrac {\binom{4}{1}+\binom{2}{1}}{\binom{5}{2}}=\frac 3 5$$

Can this be interpreted as winning in first round OR not losing in second round AND winning in third round

which would be P(wr1) + P(nlr2) * P(wr3)

P(winning in round 1) = P(red ball)/P(total) = 2/5

P( not losing in round 2) = P(picking a green ball) * P(opponent picking a green ball from remaining) = 3/5 * 2/4

P( winning in round 3) = P(picking a red ball from remaining) = 2/3

Total probability = 2/5 + (3/5 * 2/4) * 2/3

Trying to create a recurrence would be:

Pw(2,3) + Pnl(2,3) * Pw(2,1)

which is Probability of choosing red with 2 red, 3 green + Probability of not loosing * Probability of winning with 2 red and 1 green.

• Use LaTeX please. Jun 21 '19 at 4:52