First color out

We have a sack with 60 balls: 30 yellow, 20 red, and 10 white. If we draw randomly from the sack w/o replacement until all are drawn, what is the probability white is the first color with all balls removed?

Solution

Condition on a color being last ball. This reduces problem to two color case, where it is just fraction of white balls for the two colors left. If $W$ is the event of interest, and $R, Y$ are the event red and yellow are the last ball, respectively, then

\(\begin{aligned} \mathrm{Pr}(W) &= \mathrm{Pr}(W \mid Y) \mathrm{Pr}(Y) + \mathrm{Pr}(W \mid R) \mathrm{Pr}(R) \\ &= \frac{2}{3}\frac{1}{2} + \frac{3}{4}\frac{1}{3} = \frac{7}{12} \end{aligned}\)