the reflection principle
Counting the random-walk paths that touch some level — say, paths that ever reach height 5, or hit a barrier — looks hopeless directly, because there are so many ways to do it. The reflection principle is an elegant trick that turns a hard count into an easy one by a single act of mirror-flipping. It is the workhorse behind a startling number of exact random-walk results.
Here is the idea, for a symmetric walk. Suppose you want to count paths from 0 to some endpoint that touch a barrier at level a along the way. Take any such path; find the first time it hits level a; then reflect the rest of the path (everything after that first touch) across the line at height a, like folding it in a mirror. This produces a path that ends at the mirror-image endpoint, and the correspondence is exactly one-to-one and reversible. So the number of barrier-touching paths to the original endpoint equals the total number of (unrestricted) paths to the reflected endpoint — a quantity that is trivial to count. The hard, constrained count becomes a free, unconstrained one.
From this one trick a cascade of results falls out. It gives the distribution of the maximum reached by the walk up to a given time, the probability of ever crossing a level, the law of the first passage time, and the famous ballot problem (the chance one candidate leads throughout the count). Its continuous analogue does the same for Brownian motion, yielding the distribution of Brownian motion's running maximum and first-passage times in closed form. The reflection principle is a beautiful example of how a clever bijection can dissolve a difficult probability into pure counting.
To find the probability that a symmetric walk reaches level a by time n, the reflection principle relates paths that touch a to mirror-image paths, giving P(max up to time n is at least a) = P(S_n is at least a) + P(S_n is greater than a) — twice the tail (with a small adjustment), so reaching a high level is, roughly, about twice as likely as ending up there. The same fold gives the ballot result: in an election where A finally beats B by votes a versus b, the chance A leads throughout the count is exactly (a - b)/(a + b).
Reflect the path after its first touch of the barrier: the hard constrained count becomes an easy free count of mirror paths.
The reflection principle in its simplest form relies on the symmetry (each step up and down equally likely) of the walk — a clean reflection needs a process that looks the same flipped. For asymmetric walks or general processes the bijection must be adjusted or fails; do not apply the symmetric formula blindly.