Probability of picking matching socks, after partitioning the drawer.

Question

Apologies if this is a duplicate. I searched and didn't find anything quite like it.

Suppose I have a drawer with an equal number of N black socks and N white socks. They're all mixed up. So, my chances of picking a matching pair in the first two selections is (N-1)/(2N-1), right? Well, what if, before I pick the first sock, I randomly (so I don't know the colors of the socks I'm moving) partition the drawer so that there are N socks on each side, and I draw one sock from each side. Do the chances of drawing a matching pair change?

On the one hand, we can see that selection from one side doesn't change the composition of socks on the other side of the partition. However, whichever color I choose from the "first" side, it's likely that there are more of that color on that side. On other words, if I draw a black sock from one side, it's more likely that that side had N-1 blacks and 1 white than it is that that side had 1 black and N-1 whites.

My suspicion is that I need to do some kind of hypothesis testing, where I consider the chances of every possible partitioning, but that's way above my skill level.

Answer 1

Of course it shouldn't change!

Let's see if the math works out for a simple example, say $N=3$

First, the 'normal' calculation says that the chances of getting a match $M$ is:

$$P(M)= \frac{N-1}{2N-1}=\frac{2}{5}$$

Now let's see what happens when you randomly partition them.

Let's say you first pick from the left side. After randomly splitting the socks into two groups, there can be $0$, $1$, $2$, or $3$ socks there, with respective probabilities of:

$$P(0)= \frac{{3 \choose 0}\cdot{3 \choose 3}}{6 \choose 3} = \frac{1}{20}$$

$$P(1)= \frac{{3 \choose 1}\cdot{3 \choose 2}}{6 \choose 3} = \frac{9}{20}$$

$$P(2)= \frac{{3 \choose 2}\cdot{3 \choose 1}}{6 \choose 3} = \frac{9}{20}$$

$$P(3)= \frac{{3 \choose 3}\cdot{3 \choose 0}}{6 \choose 3} = \frac{1}{20}$$

Now, getting a match $M$ in the first and last situation is impossible, so $$P(M|0)=P(M|3)=0$$

When there is $1$ white sock on the left, the chances of getting a match are the chance of getting that white sock times the chances of getting one of the two socks on the right side, plus the chances of getting one of the two black ones on the left and the one black one on the right. And, with $2$ white socks on the left it's all symmetrical. So:

$$P(M|1)=P(M|2)=\frac{1}{3}\cdot \frac{2}{3} + \frac{2}{3} \cdot \frac{1}{3} = \frac{4}{9}$$

In sum:

$$P(M)=P(0)\cdot P(M|0) + P(1)\cdot P(M|1) +P(2)\cdot P(M|2) +P(3)\cdot P(M|3) =$$

$$ \frac{1}{20} \cdot 0 + \frac{9}{20} \cdot \frac{4}{9} + \frac{9}{20} \cdot \frac{4}{9} + \frac{1}{20} \cdot 0 = \frac{8}{20} = \frac{2}{5}$$

OK, so yes, same chance!

Now, I'm sure you can generalize this for any $N$ ... thus evaluating the series:

$$P(M)=\sum_{i=0}^N \frac{{N \choose i} \cdot {N \choose {N-i}}}{2N \choose N} \cdot 2 \cdot \frac{i}{N} \cdot \frac{N-i}{N}$$

[... Insert some math that's over my head ....]

... and you'll see that this ends up being $$\frac{N-1}{2N-1}$$

Answer 2

It doesn't matter if the partitioning happens before or after the first draw. Suppose it happens after. Suppose also that the first sock drawn was black.

Now, we partition off $N$ from the remaining $2N-1$, to obtain our pool for the second draw. On average, the composition of this pool is $\frac{N-1}{2N-1}$ black and $\frac{N}{2N-1}$ white. Drawing from it, we have a matching pair if we draw from the portion of it that is black, i.e., $\frac{N-1}{2N-1}$.

The trick to simplifying the work is to work with expected values for the number of socks of each color in the second part of the partition, not actual values.