A path formula for the sock sorting problem

TL;DR

This paper derives an explicit probability formula for the sequence of sock counts on a table during a sock sorting process involving pairs, providing a mathematical model for the process.

Contribution

It introduces a novel explicit formula for the probability distribution of sock counts in the sorting process, advancing understanding of this probabilistic problem.

Findings

Derived an explicit probability formula for sock count sequences

Provides a mathematical model for the sock sorting process

Enhances understanding of pairing and matching probabilities

Abstract

Suppose $n$ different pairs of socks are put in a tumble dryer. When the dryer is finished socks are taken out one by one, if a sock matches one of the socks on the sorting table both are removed, otherwise it is put on the table until its partner emerges from the dryer. We note the number of socks on the table after each of the $2n$ socks is taken from the dryer and give an explicit formula for the probability that this sequence equals a given sequence of length $2n$.