All black stones out before any white

Question

A bowl holds 3 black stones and 2 white stones, 5 in total. You draw them one at a time until the bowl is empty. What is the probability that all three black stones come out before either white stone?

Accepted Answer

Consider the whole color sequence. Picture the 5 stones in a random row of 5 positions; only which 3 positions are black matters. Every choice of "which 3 positions are black" is equally likely, and there are \binom{5}{3}=10 of them. "All blacks before any white" is exactly one pattern: blacks take the first three positions. So P=\frac{1}{\binom{5}{3}}=\frac{1}{10}=0.1. Draw-by-draw check: \frac{3}{5}	imes\frac{2}{4}	imes\frac{1}{3}=\frac{6}{60}=\frac{1}{10}. Takeaway. With smaller piles the probability climbs, but the rule is unchanged: it is 1/\binom{5}{3}, one favorable pattern over all …

All black stones out before any white

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