Probability
The law of independent trials: why past numbers can't predict the next drawing
A ball that has not appeared in 40 drawings is not "due". Here is what independence actually means, and what the full drawing record says when you test it properly.
Almost every lottery strategy ever sold rests on one assumption: that the past drawings tell you something about the next one. They do not. This is not an opinion about lotteries — it is a property of how the drawings are constructed, and it is measurable.
What "independent" means
Two events are independent when knowing the outcome of one tells you nothing about the probability of the other. Formally, P(A | B) = P(A). A lottery drawing is the textbook example: the machine is loaded with a full set of balls every time, the balls are weighed and tested, and nothing about last Friday's result changes tonight's physical setup.
The consequence is uncomfortable but simple. Before tonight's Powerball drawing, each of the 292,201,338 possible tickets has probability 1/292,201,338. The ticket 1-2-3-4-5 with PB 6 has exactly that probability. So does the combination that won last week. So does the set our own generator just produced. A weighting scheme can change which equally-likely ticket you end up holding; it cannot change the "equally likely" part.
Testing it on the full drawing record
If some numbers really were favoured, the counts would not fit a uniform distribution. That is exactly what a chi-square goodness-of-fit test measures. Below, every drawing since each game's current ball matrix began — 2,312 drawings in total — is tested against the hypothesis that every ball is equally likely.
| Mega Millions | Powerball | |
|---|---|---|
| Drawings tested | 920 | 1,392 |
| White balls | 1–70 | 1–69 |
| Expected appearances per ball | 65.7 | 100.9 |
| Chi-square statistic | 60.3 | 78.0 |
| Degrees of freedom | 69 | 68 |
| p-value | 0.76 | 0.19 |
Under a fair game the chi-square statistic should land near its degrees of freedom (69 and 68 respectively), and it does. The p-values — 0.76 and 0.19 — are the probability of seeing a spread of counts at least this uneven purely by chance. Neither is anywhere near the conventional 0.05 threshold. There is no statistical evidence of bias in either game.
But some numbers have come up far more often
They have, and that is the point. In 1,392 Powerball drawings, ball 21 has appeared 124 times while ball 13 has appeared only 76 times. A gap of 48 appearances looks like a signal. It is not.
To show why, we simulated 4,000 alternative histories of the same length using a uniform random generator — a game guaranteed to be fair — and recorded how extreme the most- and least-drawn balls were each time.
| Real record | Fair simulation (average) | |
|---|---|---|
| Mega Millions: most-drawn ball | 85 (ball 10) | 85.1 |
| Mega Millions: least-drawn ball | 51 (ball 51) | 47.8 |
| Mega Millions: hottest-to-coldest gap | 34 | 37.3 |
| Powerball: most-drawn ball | 124 (ball 21) | 124.6 |
| Powerball: least-drawn ball | 76 (ball 13) | 78.5 |
| Powerball: hottest-to-coldest gap | 48 | 46.1 |
The real "hot" and "cold" extremes sit essentially on top of what a provably fair machine produces. Spread of that size is not evidence of bias; it is the expected consequence of distributing a few thousand appearances across seventy independent slots. If the counts came out perfectly level, that would be the anomaly worth investigating.
Three things people expect to see, and what actually happens
1. "A number is due after a long absence"
The longest current dry spell in Mega Millions belongs to ball 8, last seen 49 drawings ago. Its probability of appearing tonight is 5/70 — identical to every other ball, and identical to what it was the day after it last appeared. Dry spells of that length are ordinary: with 70 balls and 5 drawn per game, the average wait between appearances is 14 drawings, and waits several times longer than average are routine in any memoryless process.
2. "Numbers from the last drawing won't repeat"
In 920 Mega Millions drawings, 32.8% contained at least one ball from the immediately preceding drawing; for Powerball it is 31.9%. Repeats are not rare — they are the norm in roughly a third of drawings, exactly as independence predicts.
3. "Consecutive numbers never come up"
25.8% of Mega Millions drawings and 26.4% of Powerball drawings contain at least one pair of consecutive numbers. Avoiding them on purpose removes about a quarter of the real winning patterns from your ticket for no reason at all.
The one thing your choice does affect
Independence kills prediction, but it does not make every ticket equally valuable. If you win a jackpot, you split it with everyone else holding the same combination — and human number choices are extremely predictable. Dates, sequences and patterns on the play slip are picked far more often than the pool would suggest. Choosing unpopular combinations cannot improve your chance of winning, but it can improve what you keep if you do. That is the only defensible edge in number selection, and it is about how other players pick, not how the balls fall.