# The probability to match pair of socks [closed]

There is $n$ pairs of socks that each sock having one mate,each pair is different from another pair .

There is $2$ drawers: left drawer and right drawer.

All the left socks are in the left drawer and all the right socks are in right drawer.

Every day I take $1$ sock from the left drawer and $1$ sock from the right drawer.

The socks are worn at that day and then thrown into a laundry basket,so after $n$ days the drawers will be empty.

What is the probability that in day number $k$ , I took a matching pair,when $1\le k \le n$ ?

I want to find the probability that a match occurs on day $k$?

How do I approach to that type of question?

## closed as off-topic by Did, Shailesh, C. Falcon, Zain Patel, Claude LeiboviciApr 23 '17 at 7:18

This question appears to be off-topic. The users who voted to close gave this specific reason:

• "This question is missing context or other details: Please improve the question by providing additional context, which ideally includes your thoughts on the problem and any attempts you have made to solve it. This information helps others identify where you have difficulties and helps them write answers appropriate to your experience level." – Did, Shailesh, C. Falcon, Zain Patel, Claude Leibovici
If this question can be reworded to fit the rules in the help center, please edit the question.

• Are you asking for the probability that the FIRST match occurs on day $k$ or just the probability that a match occurs on day $k$? Also, are you replacing the socks each day? – lulu Apr 18 '17 at 12:33

## 2 Answers

The chance you get a match on day $1$ is $\frac 1n$. Imagine you pull from the left first, then there are $n$ socks in the right drawer to choose from and $1$ matches.

The chance for any day is also $\frac 1n$. Imagine lining up all the socks in the order you will pull them. For a given day $k$, imagine swapping the first and $k$th stocks in both rows. Now day $k$ is a match if and only if the first day would have been a match before the swap. It is just confusing yourself to worry about whether day $1$ matched before you draw day $k$. You can do that, but it will still come out $\frac 1n$.

• But after $k$ days,I left with $n-k$ socks in each drawer. Why I don't need to check it? – Asaf Apr 21 '17 at 15:45
• I showed why by the swap argument. The computation is not hard, either. On day $k$ you pull left sock. There is $\fraf{k-1}n$ chance you have pulled its mate, so $\frac {n-k}n$ its mate is available and $\frac 1{n-k}$ chance you pull it . Multiply them and you get $\frac 1n} – Ross Millikan Apr 21 '17 at 16:28 Let's number the socks. We call them$\ell_1,\ell_2,\ldots, \ell_n$for the left ones and$r_1,r_2,\ldots, r_n$for the right ones. Now you pick one of each, so the set of possible outcomes is$\{(\ell_i,r_j) \mid 1 \leq i,j\leq n\}$, a set with$n^2$elements. To take a matching pair, we have to take from the subset$\{(\ell_i,r_i) \mid 1 \leq i \leq n \}$, a set with$n$elements. As we are assuming that every pair has the same probability, can you compute the chance of picking a matching pair (not considering$k$right now)? Let's for now call this probability$p$. The second part of your questions asks: If we repeat the above experiment with chance to succeed$p$every time, what is the chance that we first succeed after$k$tries? Sounds familiar? (Hint: Bernoulli...) • depends if you put the socks back in to the drawers or not – JJR Apr 18 '17 at 12:32 • Also the OP does not specify that the match on day$k$be the first one. – lulu Apr 18 '17 at 12:32 • And he also did not mention that all socks can be distinguished, so yes, I made some assumptions... :) – Dirk Apr 18 '17 at 12:34 • @lulu - ignoring matches or not on other days, the probability of a match on day$k$is the same as on day$1$. – Henry Apr 21 '17 at 15:17 • @Henry Yes, but the probability of getting the first match on day$k$is different than the probability of getting a match on day$k\$. – lulu Apr 21 '17 at 15:18