A drawer has 18 black socks and 14 white socks. On average, how many socks should one draw to get two equal socks? And to get two different socks?
Solution A
The answer to the first one is trivial: 3. For the second question, let $N = 18$ be the number of black socks and $M = 14$ the number of white socks. Then:
| outcome | probability |
|---|---|
| $bw$ | $\frac{N}{N+M} \frac{M-1}{N+M+1}$ |
| $wb$ | $\frac{M}{N+M} \frac{N-1}{N+M+1}$ |
| $bbw$ | $\frac{N}{N+M} \frac{N-1}{N+M-1} \frac{M-1}{N+M}$ |
| $wwb$ | $\frac{M}{N+M} \frac{M-1}{N+M-1} \frac{N-1}{N+M}$ |
| $bbbw$ | $\frac{N}{N+M} \frac{N-1}{N+M-1} \frac{N-2}{N+M-2} \frac{M-1}{N+M-1}$ |
| $wwwb$ | $\frac{M}{N+M} \frac{M-1}{N+M-1} \frac{M-2}{N+M-2} \frac{N-1}{N+M-1}$ |
| … | … |
Due to linearity of expectation, the expected value is, by definition:
\[\sum_{k=1}^N{(k + 1)\left(\prod_{i=1}^k{\frac{N + 1 - i}{N + M + 1 - i}}\right)\frac{M}{N + M - k}} + \sum_{k=1}^M{(k + 1)\prod_{i=1}^k{\frac{M + 1 - i}{N + M + 1 - i}}\frac{N}{N + M - k}}\]Which, for the given values of $N$ and $M$, gives a result of $^{279}/_{95}$, or 2.93684.
Solution B
The definition of an negative hypergeometric distribution is:
(…) the probability of the number of elements taken without replacement from a finite population whose elements can be classified into two mutually exclusive categories like Pass/Fail, Male/Female or Employed/Unemployed that stops when a fixed number of elements of certain class have been taken
If there are $N = 18 + 14 = 32$ elements, of which $K = 18$ are defined as “successes” and the rest are “failures”, and the elements are drawn one after the other, without replacements, until $R = 1$ failures are encountered, we know that the expected value is given by:
\[f(N,K,R) = \mathbb{E}[x \sim NHG_{N,K,R}] = R ~ \frac{K}{N-K-1}\]Consider two cases: one in which the first ball drawn is black \(p =~^{18}/_{20}\) and the other in which the first ball is white \(q =~^2/_{20}\). You can then calculate your average for each case and weight them as follows:
\[\frac{18}{20} ~ f(19, 17, 1) + \frac{2}{20} ~ f(19, 1, 1) + 2\]The last term refers to the first draw (which is considered separately) and the last “failure” draw, which is not counted by the negative hypergeometric distribution.