Flip a coin, step left or right, and discover why gamblers always go broke
Put a marker at zero, flip a fair coin, and move one step right for heads or left for tails. Keep going forever and the marker will revisit every point on the line infinitely often. That simple game, a random walk, also explains why a gambler with finite money must eventually lose to a bank with unlimited funds.
A random walk is a path built from a sequence of random steps. Karl Pearson introduced the term in 1905, and some texts call it the drunkard's walk. The same model describes a molecule jostling through a liquid or gas, an animal searching for food, a fluctuating share price and a gambler's bankroll, which is why it turns up in physics, ecology, economics, computer science and psychology.
The coin version shows the arithmetic. After five flips the marker can only sit at an odd position between minus five and five. Landing on one needs three heads and two tails in any order, which can happen in ten ways, while reaching five requires five heads in a row, just one way. These counts come straight from Pascal's triangle, and as the number of flips grows, the pattern smooths into the familiar bell-shaped normal curve.
On average the walker goes nowhere: the expected position stays at zero, because every step is equally likely to go either way. Yet the typical distance from the start keeps growing, roughly in proportion to the square root of the number of steps. There are neat exact results too. Starting at zero, the expected number of steps before first reaching either b or minus a is simply a times b.
The most striking property is recurrence. A simple walk on a line will cross every point infinitely many times if allowed to continue, a result also known as the level-crossing phenomenon or gambler's ruin. In a fair game against an opponent with limitless money, the gambler's fortune wanders at random and, sooner or later, it touches zero and the game ends.
Source: Random walk
