Reflection Principle

The reflection principle is a technique for counting lattice paths that cross (or avoid) a boundary. Setup: Count paths from $(0,a)$ to $(n,b)$ that touch or cross $y = 0$. Reflect the portion of the path before the first crossing about $y = 0$. This creates a bijection with all paths from $(0,-a)$ to $(n,b)$. Result: The number of "bad" paths (those that cross the boundary) equals the total paths from the reflected starting point. The number of "good" paths (those that stay above the boundary) is the difference. Ballot problem: If candidate A gets $a$ votes and B gets $b$ votes with $a > b$, the probability that A is strictly ahead throughout the count is $\frac{a-b}{a+b}$. When to use: Counting lattice paths that avoid a boundary. Derivation of the Catalan number formula. Any problem where you need to count paths constrained to stay on one side of a line. This is a fundamental technique. The Catalan number formula, the ballot problem, and many random walk results follow from the reflection principle.