The probability to match pair of socks 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?

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 $\frac 1n$.