the arcsine law
Here is one of the most counterintuitive results in all of probability. Two equally skilled players gamble on a fair coin for a long time, tracking who is ahead. Common sense says the lead should change hands often and each player should be ahead about half the time. Common sense is badly wrong. The arcsine law reveals that the time one player spends in the lead is most likely to be near 0 percent or near 100 percent of the game — and least likely to be near the 'fair' 50 percent.
Make it precise. For a symmetric random walk (or Brownian motion) run over a long stretch, consider the FRACTION of time the path spends on the positive side. You might expect that fraction to cluster around 1/2. Instead its distribution is the arcsine distribution, which is U-shaped — piled up at both ends near 0 and 1, with a dip in the middle. The probability that the fraction of time spent positive is at most x is given by (2/pi) times arcsin(sqrt(x)), hence the name. The most likely outcomes are that one side dominates almost the whole time; a near-even split is the rarest case of all.
The intuition behind the paradox is subtle but real: once a random walk drifts to one side, returning to and crossing zero is hard and slow (recall that the expected time to return to the origin is infinite). So the path tends to stay on whichever side it has wandered to, for long stretches at a time. Three sibling arcsine laws govern this regime — the fraction of time positive, the time of the last visit to zero, and the time of the maximum — all share the same U-shaped distribution. The practical moral is stark: in a long fair game, long one-sided 'streaks' are not flukes; they are the norm. Reading them as evidence of skill, momentum, or a rigged game is a classic statistical error.
Flip a fair coin 10,000 times, scoring +1 for heads and -1 for tails, and ask: for what fraction of the running total's life was it positive? The arcsine law says you are FAR more likely to see something like 'positive 96 percent of the time' or 'positive 3 percent of the time' than the seemingly natural 'positive 50 percent of the time'. The probability the positive fraction lands in the narrow band around 1/2 is the smallest of all.
The fraction of time a fair walk spends ahead has a U-shaped (arcsine) law — near-total dominance is common, an even split rarest.
The arcsine law is a sharp warning against the gambler's-fallacy intuition that a fair game must 'even out' moment to moment. It does not: long, lopsided leads are the typical behaviour of a fair random walk, not a sign of bias or skill.