# The Construction of Two Kinds of Bijections in Simple Random Walk Paths

## TL;DR

This paper constructs two natural and simple bijections between sets of paths in 2n-step symmetric simple random walks, enabling proofs of equal probabilities for certain events and facilitating further analysis.

## Contribution

It introduces two novel bijections between path sets with equal cardinality, providing a new method to analyze probabilities in symmetric simple random walks.

## Key findings

- Bijections are natural and easy to implement.
- The construction helps prove equal probability of events.
- Facilitates further probabilistic analysis in random walks.

## Abstract

It is known that for the 2n-step symmetric simple random walk on Z, two events have the same probability if and only if their sets of paths have the same cardinality. In this article, we construct two kinds of bijections between sets of paths with the same cardinality. The construction is natural and simple. It can be easily realized through programming. More importantly, this construction opens a door to prove that two events in the 2n-step symmetric simple random walk on Z have the same probability and some further related results.