Pair of Socks Puzzle
Probability
Level
3
A drawer contains 12 identical black socks and 12 identical white socks. If you pick 2 socks at random, what is the probability (in %) of getting a matching pair?
Round your answer to the nearest tenth place in the decimal.
The answer is 47.8.
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1 solution
The tempting answer is 50 percent. But this is wrong! Surprisingly, the answer is less than 50 percent. It is actually 11/23, which is about 47.8 percent and slightly less than half. Here is the solution : By symmetry, the probability of making a black pair is the same as the probability of making a white pair. What’s the chance of making a black pair? For the first sock, there are 12 black socks out of 24 total socks for a 12/24 chance. After we remove 1 black sock there are 23 remaining socks of which 11 are black. The chance of picking a second black sock is 11/23. The chance of picking two black socks is then (12/24)(11/23) = (1/2)(11/23). The slight asymmetry of 11/23 is because we are sampling without replacement. The probability of selecting a pair of white socks is also (1/2)(11/23). The overall chance of making a pair is then:
Pr(pair) = Pr(black pair) + Pr(white pair)
Pr(pair) = (1/2)(11/23) + (1/2)(11/23)
Pr(pair) = 11/23
We can generalize to 2n black socks and 2n white socks with a similar calculation.
The first sock has a 2n/(4n) = 1/2 chance of being a particular color, and the second sock has a (2n – 1)/(4n – 1) chance of being the same color as the first sock.
Pr(pair) = Pr(black pair) + Pr(white pair)
Pr(pair) = (1/2)(2n – 1)/(4n – 1) + (1/2)(2n – 1)/(4n – 1)
Pr(pair) = (1/2)(2n – 1)/(4n – 1)
The above fraction is less than 1/2 for positive integers n and it approaches 1/2 as n goes to infinity.
11/23 is about 47.8%.