Two friends flip a fair coin a thousand times. Heads, she scores a point; tails, he does. Keep a running score, and ask a simple question: how much of the game was she ahead? Half, you'd think, give or take. Instead, in game after game, one of them leads almost the whole way, and the mathematics says the lopsided game is not the exception. It is the most likely thing a fair coin can do.
For the curious kid.
Here is the trick of it. Once one player gets a few points ahead, the coin does nothing to bring her back: it has no memory and no sense of fairness. It just keeps flipping. To lose the lead, the other player has to make up the whole gap by luck alone, and the further ahead she is, the longer that takes. So leads tend to last. In a thousand flips, the lead changes hands only about a dozen times, and in one game out of twenty one player is never behind for a single flip. Play a game and look at the line: long stretches all gold, or all blue, and only a few crossings. Then play a thousand, and look at the histogram: the pile is deepest at the two ends (games where one side led nearly always) and shallowest in the middle, where a "fair-looking" game would be.
This matters outside of coin games. A basketball player who hits six shots in a row, a fund manager who beats the market five years running, a friend who seems to win every argument: before you call it a hot hand, ask what pure chance would look like. It looks like this. Streaks and long leads are what fairness produces, not what it prevents.
Deeper.
The running score is a simple random walk, and the fraction of time it spends above zero obeys the arcsine law: as the number of flips grows, the probability that one side leads for less than a fraction x of the game approaches (2/π)·arcsin √x. Its density, 1/(π√(x(1−x))), is the gold U-curve: infinite at the ends, lowest at the middle. Paul Lévy found it for Brownian motion in 1939; William Feller made it the centrepiece of his great textbook, where he wrote that the results are "so amazing and so at variance with common intuition" that even his sophisticated colleagues doubted coins really behave this way. For a finite game of 2m flips the law is exact and short: the probability that one side leads for exactly 2k flips is u2k·u2m−2k, where u2j = C(2j, j)/4j is the probability that a walk of 2j steps ends at zero. That exact law is the blue staircase over the bars.
The second histogram is a twin surprise: the moment of the last tie (the last time the score was level) follows the very same law. The last tie is most likely to be near the very start or the very end, and least likely in the middle. Both facts come from one idea, the reflection principle: for every walk that touches zero and then rises, there is a mirror-image walk that touches zero and then falls, so paths that return to zero are exactly as numerous as paths that end at zero, and the probability of no return in 2m steps is u2m ≈ 1/√(πm), shrinking only as the square root. That slow shrinking is why the lead is so sticky.
Honesty notes, museum policy. The coin is an honest pseudo-random flip. "Time in lead" follows Feller's convention: a flip counts for the side that is ahead before or after it (the score cannot be tied on both ends of a flip), and the blue staircase is the exact finite-game law, never fitted; the gold curve is the arcsine limit, also never fitted, and at a thousand flips it sits within 2% of the exact law on every bar. The laws are computed from the tie probabilities u2j alone, and the whole thing was checked headless against an exhaustive enumeration of every one of the 1,048,576 twenty-flip games, where the counts match to the last integer.
The coin is fair. That is exactly why the game is not.