Six things commentators say about momentum, checked against every point of 11,452 charted professional matches — and then checked again against tennis played by coins, where momentum cannot exist. The coins say the same thing.
Every one of these is said out loud during matches, and every one of them is true of the data as stated. Pick one — the rest of the page follows whichever you choose.
The number under each claim is the plain comparison: the situation against everything else, with no adjustment at all. They are not wrong. The question is what else would produce them.
The obvious objection to the plain comparison is that the situations are not handed out at random. A player who is serving for the set is usually the better player, having a good day, against someone having a bad one. So compare like with like: hold the player fixed and compare them with themselves — across their career, then inside a single match, then inside a single set.
The answer does not settle down. It moves, and for serving for the set it changes sign: from +4.3 per 100 games with no control to −14.8 with the tightest one. Tightening a control is supposed to remove a bias, not manufacture one. So which of these four numbers is the answer?
None of the four numbers is the answer, because a control can only tell you what it is measuring if you know what it says when there is nothing to find. So here is nothing to find: every match played again as independent coin flips. Each simulated match keeps its two real players, its format, and two serve strengths taken from what those players actually did — and has no memory whatsoever. A point never knows what the last point did.
The serve strengths are shrunk toward each player's career rate by 30 points of prior, which is not a free choice: it is the value that makes the simulated matches produce the real distribution of scorelines. Deciding sets come out at 30.9% against 30.0% real, tiebreaks 17.9% against 16.2%, holds 76.0% against 75.8%.
Each run replays all 11,452 matches in your browser and re-runs the identical test on the result. The green line is the real data. Nothing in the simulator knows what a streak is.
All six claims, each at its tightest honest control, as a difference from what memoryless tennis produces under the identical test. Zero means the claim has no content beyond the shape of the situation it describes. Every interval crosses zero.
Units are the excess per 100 trials — per 100 service games for the first four, per 100 points for the hot hand, per 100 deciding sets for the last. Bars are 95% intervals combining the real sample's own error with the Monte-Carlo error of the control.
The fifth claim is the famous one, and it is the one that nearly survives. Players do win 65.0% of the points after a point they won, against 63.0% after a point they lost. Control for the player, the match, the set, and a gap is still there — small, but bigger than the coins produce.
It dies on a detail of scoring. Two points played at 30–15 have had exactly the same number of points won and lost; the only difference is the order they came in. So compare points at the identical score, and the comparison is purely about sequence — nothing else can differ.
Two wins and one loss either way. Under that control the hot hand is +0.04 per 100 points against the coins' own ±0.11 — gone. What the looser controls were picking up was not sequence but score: servers play 40–0 differently from 0–40, and "won the last point" is partly just a way of saying "is ahead in this game".
Adding the score control halves the usable points — at 15–0 or 0–40 there is only one way to have got there, so those points say nothing about order and drop out of the comparison. 881,838 points become 324,581.
And it is not hiding in a subgroup: men, women, all three surfaces, both formats, three eras. Every one is a null under the score control.
A null is only as good as its power, so here is the power. Put a real dependence back into the simulator — after a point the server won, raise their chance by δ; after one they lost, lower it so the long-run rate is untouched — and see how much of it the test recovers. The relationship is linear, and the interval above excludes anything bigger than 0.10 percentage points per point. One point in a thousand.
The grey band is what the test would have detected; the dashed line is the largest dependence consistent with the observed data.
And even if it were there, it would be close to invisible. A dependence of 0.015 — fifteen times the upper bound — moves the share of matches won by the weaker server from 17.3% to 18.3%, and deciding sets from 30.8% to 32.1%. A whole season of tennis would look almost exactly the same.
This is not a sample of professional tennis. It is a sample of the matches dozens of volunteers chose to sit down and chart, point by point, and they chose the matches people want to watch: finals, Grand Slams, and the very best players. Whether that biases a within-match comparison is a fair question, and the honest answer is that it could — if momentum were real only in the matches nobody charts, this page would not find it.