  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.
