Socks in the Dark

Question

A drawer contains three pairs of socks, each pair a different color, six socks in all. You pull socks out one at a time in the dark. What is the probability you need exactly three draws to obtain a matching pair?

Accepted Answer

Multiply the two required conditional probabilities. First two differ. Whatever the first sock is, 4 of the remaining 5 socks are a different color: P(textno pair after 2) = frac45. Third completes a pair. You now hold two different colors; the drawer has 4 socks left, the two mates of what you hold, plus the untouched third pair. So P(textthird matches mid textfirst two differ) = frac24 = frac12. P(textexactly 3 draws) = frac45 cdot frac12 = frac25 = 0.4. Sanity check, the full distribution P(2 text draws) = frac15, P(3) = frac25, and by the pigeonhole principle four draws from three colors must contain a pair, so P(4) = 1 - frac15 - frac25 = frac25. The distribution sums to 1, and the certainty at four draws is a free consistency check. The reusable skill here is chaining conditional probabilities while updating the population: exactly the bookkeeping that without-replacement card and inventory problems demand. Sloppy candidates forget that "two mates" is the count of useful socks, not the count of colors.

Socks in the Dark

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