How the odds got longer

The odds, and the wait they imply

Jackpot odds. For a game drawing $k$ numbers from $N$, the number of distinct tickets is the binomial coefficient

$\binom{N}{k}=\frac{N!}{k!\,(N-k)!}.$

Where a bonus ball comes from its own machine, the two choices are independent and the counts multiply:

$\text{odds}=\binom{N}{k}\times\binom{B}{b}.$

Where the bonus is drawn from the same drum as the main numbers, it does not multiply anything — it distinguishes prize tiers below the jackpot, not the jackpot itself. Getting that distinction wrong turns Canada's 6/49 into one chance in 685 million instead of one in 13,983,816, and the flag in the database that is supposed to record it is wrong on 55 formats, so the site decides it from the pool ranges instead.

The sum scale. For $k$ numbers drawn without replacement from $1\ldots N$, the sum has

$\mu=\frac{k(N+1)}{2},\qquad \sigma=\sqrt{\frac{k(N+1)(N-k)}{12}}.$

Both follow from the mean and variance of a single draw and the covariance induced by drawing without replacement. They move with every rule change, which is why a "sum range" carried over from one era to the next is meaningless.

The wait between jackpots. With $S$ lines in play and one in $\Omega$ of them winning, the number of winners is very nearly Poisson with mean $S/\Omega$, so the chance at least one ticket wins is

$p=1-e^{-S/\Omega}.$

Draws pass without a winner independently, so the count of misses before a hit is geometric and its mean is

$E[\text{run}]=\frac{1}{p}-1.$

This depends on how many people played, not only on the rules, which is why the column is empty for eras with no sales figure. It is the pre-draw expectation; the observed column beside it is what actually happened.

The observed wait. Counted only over runs closed at both ends. A run still open when an era finished has no length yet, and counting it as though it had would bias every figure downward — the direction that makes a game look easier to win than it is.

What the observed column will not do. Two readings are refused outright: an era whose top prize was won on every draw, and one where it was never won, over thirty draws or more. Neither describes a lottery. Both occur in the archive, and both are prize tables that do not carry the jackpot row rather than facts about the game.

The redesign nobody announces in full

On 10 October 2015 the UK's national lottery drew six numbers out of 59 for the first time. Until the previous week it had drawn six out of 49, as it had since 1994.

The announcement talked about a new Millionaire Raffle, a bigger prize for matching two numbers, and rolling jackpots that would climb faster. All of that was true. What it did not put in a headline is the arithmetic: six from 49 is one chance in 13,983,816, and six from 59 is one chance in 45,057,474. Overnight, the jackpot became 3.22 times harder to win.

Powerball did the same thing that year, four days earlier. Its pool went from 59 white balls to 69 while the powerball pool shrank from 35 to 26 — a combination that reads as generous on one side and is not: one in 175,223,510 became one in 292,201,338.

Both changes did exactly what they were designed to do. A jackpot won less often rolls over more often, so it grows larger between wins; a larger advertised jackpot sells more tickets; and by January 2016 Powerball had produced a $1.586 billion prize that no 5-from-59 game would have reached. The operators were not hiding anything. The number is in the published game rules, and anyone can compute it.

That is what this page exists to do. Not to accuse — a lottery is entitled to redesign its game, and the tradeoff between "won often, smaller" and "won rarely, enormous" is a genuine design choice with players on both sides of it. But the size of the change is rarely in the announcement, and the arithmetic that gives it is four lines long.

One column is worth reading alongside the odds: the wait. Powerball's current era has gone an average of 17 draws between jackpot winners, against 7 draws in the era before the 2015 redesign. That is the change, measured in the only unit a player actually experiences — the number of Wednesdays and Saturdays that pass with nobody winning.