# Pairing random socks

**December 27, 2020.** *If you have a jumbled pile of socks,
how many do you need to draw on average before getting a pair? The
answer turns out to be surprisingly tricky!*

#### Searching for socks

After a load of washing, I sometimes get lazy and throw unpaired socks into a draw. Later, I will simply withdraw socks at random until I get a pair. Faced with this dilemma, I wondered: for $n$ pairs, what is the average number of draws required to get a pair? This simple question turns out to be surprisingly tricky to answer!

To start with, we can calculate the probability that after $k$ draws, you have no pair. This will simply be the number of ways of drawing $k$ socks with no pairs divided by the total number of ways to draw $k$ socks from the total number, $2n$. Since we have to choose $k$ distinct pairs to draw from, and within each pair there are two options, the number of ways to draw $k$ sock without a pair is

\[2^k k! \binom{n}{k} = \frac{2^k n!}{(n-k)!}.\]Thus, the probability that no pairs are drawn is

\[p_k = \frac{2^k n!}{(n-k)!} \cdot \binom{2n}{k}^{-1} = \frac{(n!)^2}{ (2n)!} \cdot 2^k\binom{2n-k}{n-k}.\]We have tried to simplify the $k$ dependence so there is a single binomial coefficient. Note that $p_k = 0$ for $k \geq n + 1$. This reflects the fact that as you soon as you have drawn more than half the socks, you are guaranteed a pair by the pigeonhole principle. Let $D$ be the number of draws needed to get a pair. Then

\[p_k = \Pr(D > k).\]We can use this to find the probability that we get a pair after exactly $k$ draws:

\[\Pr(D = k) = \Pr(D > k - 1) - \Pr(D > k + 1) = p_{k-1} - p_{k+1}.\]We can evaluate this more explicitly, but this will end up being a distraction from our main goal: to compute the average number of draws to get a pair. Since we have an expression for the probabilities $\Pr(D = k)$, we can go ahead and compute the average:

\[\langle D\rangle = \sum_{k = 0}^n k \cdot \Pr(D = k) = \sum_{k = 0}^n k \cdot (p_{k-1} - p_{k+1}). \tag{1} \label{sum}\]#### A hypergeometric hack

To make progress on this sum, we can use a trick. We note that each term $p_k$ occurs twice, first with a multiplier $k+1$, then a multiplier $-(k+2)$. Combined, each term appears precisely once! Thus, we can simplify the sum to

\[\langle D\rangle = D_n = \sum_{k = 0}^n p_k = \sum_{k = 0}^n \frac{(n!)^2}{ (2n)!} \cdot 2^k\binom{2n-k}{n-k}.\]This is a difficult sum, and there is (as far as I know) no closed
form expression in terms of elementary functions.
Instead, we can invoke a special function called the *Gauss
hypergeometric function* to package things nicely.
As nicely described in
*Concrete Mathematics*,
the hypergeometric function captures any sum whose terms differ by a
rational function of $k$.
More precisely, the rule is that if we have terms $t_k$ for $k \geq
0$, with a ratio

then we can package the sum of the terms into a hypergeometric function:

\[\sum_{k\geq 0} t_k = t_0 \cdot {}_2 F_1(a, b; c; z).\]There is a more general version of this relation which lets us package
things into the *generalized hypergeometric function* but we won’t
need that here.
Let’s apply the relation above to the randomly drawn socks $(\ref{sum})$.
The ratio of the terms $p_k$ is (after a little algebra) seen to be

Note that $p_0 = 1$, since it is certain you cannot draw a pair after drawing a single sock. Thus, the average number of socks you need to randomly draw from a pair from $n$ jumbled pairs is exactly

\[\langle D\rangle = {}_2 F_1 (-n, 1; -2n; 2). \label{hyper} \tag{2}\]#### Simulating socks

As a sanity check, we can simulate the random drawing of socks and see that the answers agree with our formula. Here is a scatterplot of the data for $n = 1$ to $n = 20$ pairs of jumbled socks, with red datapoints obtained from simulations, and blue from the analytic expression $(\ref{hyper})$. It’s a pretty good match, and gets better as you increase the number of simulations:

If you’re interested, here is the Python code that generates this plot:

```
import numpy as np
import scipy.special as sc
import matplotlib.pyplot as plt
def randsocks(n):
socks0 = [[i, 0] for i in range(n)]
socks1 = [[i, 1] for i in range(n)]
rand = np.random.permutation(socks0+socks1)
firstrand = [i for [i, j] in rand]
k = 0
while len(firstrand[:k]) == len(list(set(firstrand[:k]))):
k = k + 1
return k
averages = []
repeats = 1000
for n in range(1,21):
results = []
for rep in range(repeats):
results.append(randsocks(n))
av = sum(results)/float(len(results))
averages.append(av)
hyper = []
for n in range(1, 21):
hyper.append(sc.hyp2f1(1, -n, -2*n, 2))
fig=plt.figure()
ax=fig.add_axes([0,0,1,1])
ax.scatter(range(1,21), averages,color='r')
ax.scatter(range(1,21), hyper,color='b')
plt.show()
```

#### Alien feet

Lazy aliens might encounter the same problem I did, but with a difference: they have more than two feet! In this case, our answer above generalizes in a nice way. If the alien has $f$ feet, and draws with replacement from $n$ sets of $f$ socks, then the probability that it has no sets after $k$ draws is

\[p_k = \frac{(n!)^2}{(fn)!} \cdot f^k\binom{fn-k}{n-k}.\]The same trick can be used to evaluate the expected number of draws, $\langle D\rangle$, and we end up simply replacing $2$ with $f$ in our expression $(\ref{hyper})$:

\[\langle D \rangle = \sum_{k=0}^n p_k = {}_2 F_1 (-n, 1; -fn; f).\]We can check numerically that this increases as $f$ gets bigger, just as you might expect. Pluripedality may have its advantages, but when it comes to drawing socks from a giant disorganized pile, monopods have a leg up.

#### Biographical postscript

When I was 10 or so, my older sisters gave me a copy of
*Math Curse*, a picture
book about someone who sees math problems everywhere.
Clearly, I have fulfilled their prediction and become a math zombie!