The construction of two kinds of bijections in simple random walk paths

It is known that for the 2 n-step symmetric simple random walk on \mathbbZ, 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 2 n-step symmetric simple random walk on \mathbbZ have the same probability and some further related results.