The odds of winning the lottery twice (and why it keeps happening)

Every year or two, a headline announces that someone has won the lottery again, and a journalist multiplies two odds together to produce a number with thirteen zeroes. The multiplication is correct. The conclusion — "a one-in-quadrillions miracle walks among us" — is not. Both facts fit on one page, and the gap between them is the most instructive piece of probability in the whole lottery world.

The squared odds are real — for a question nobody asked

Probabilities of independent events multiply. So the chance that a specific person, buying one line in each of two specific draws of Saturday Lotto (jackpot odds 1 in 8,145,060 per line, derived in how lottery odds are calculated), wins Division 1 both times is:

(1 ÷ 8,145,060) × (1 ÷ 8,145,060) = 1 ÷ 66,342,002,403,600

One in 66.3 trillion. For two named draws of US Powerball it is (1 ÷ 292,201,338)² = 1 in 85,381,621,928,990,244 — about 1 in 85 quadrillion. If you pointed at a stranger and two future draw dates and said "her, those two nights", that is the bet you'd be making.

But that is never the event in the news story. The news story is: someone, somewhere, among all the people who play, won twice, in any two draws of any game over any span of years. Different question, different arithmetic, wildly different answer.

The law of truly large numbers

Statisticians Persi Diaconis and Frederick Mosteller, in their classic 1989 paper on coincidences, put it bluntly: "with a large enough sample, any outrageous thing is likely to happen" (Methods for Studying Coincidences, JASA 1989). Their worked example was precisely a double lottery winner. Discussing a New Jersey woman's second jackpot — reported at the time as a 1-in-17-trillion event — they cite Purdue statisticians Stephen Samuels and George McCabe, who reframed the question as "some past winner, among the millions of US players, hits another jackpot in some draw" and found the odds were about 1 in 30 over a four-month window, and better than even over seven years.

The repair is the same one as in the birthday problem: count the attempts, not the person in the headline. Past jackpot winners keep playing — often heavily, since they now have both the habit and the bankroll. Thousands of past winners × several draws a week × multiple games × decades: the universe gets billions of attempts at manufacturing a double winner. It would be genuinely strange if double winners didn't keep appearing.

Real double winners, on the record

Evelyn Adams — New Jersey, 1985 and 1986. Adams won a $3.9 million New Jersey Lottery jackpot in October 1985 and a $1.4 million jackpot that February — $5.4 million across two wins a few months apart, reported at the time by The New York Times under the headline "Odds-Defying Jersey Woman Hits Lottery Jackpot 2d Time". Hers is the case Diaconis and Mosteller used. Two footnotes make it even less miraculous than the squared-odds figure suggests: she played regularly (many attempts, not two), and the games' individual odds were millions-to-one, not hundreds-of-millions. Her later life — the winnings were gone within years (Wikipedia) — belongs to another cluster of stories entirely.

Bill Morgan — Australia, 1999. A Victorian truck driver who had recently survived a heart attack and a 14-minute clinical death won a car worth about A$30,000 on a scratchcard. Nine News asked him to re-enact the scratch for their cameras two weeks later; on camera, the replacement ticket won him A$250,000 (Wikipedia — Bill Morgan lottery win). It remains the best-filmed illustration of the law of truly large numbers in existence: of all the thousands of feel-good re-enactments television has ever staged with scratchcards, one of them was always eventually going to win on air — and that's the one you've seen.

How to read the next "won it twice" headline

1. Check which question the quoted odds answer. A number with 13+ zeroes is the specific-person, specific-draws squared calculation. It is almost never the odds of the event that actually occurred.
2. Count the exposure. Multiply players by draws by years. Big-number events need big-number denominators, and lotteries supply them for free.
3. Remember the asymmetry. "Someone wins twice" is near-certain across a population; you winning even once remains 1 in 292,201,338 per Powerball line (official odds). The headline changes nothing about your ticket — the same trap as someone always wins, one level up.

Double winners are not evidence that lightning aims. They are evidence that there is a great deal of lightning. You can watch the effect emerge yourself by running enough trials in the draw simulator — given enough simulated players, the double winner always shows up, and it is never you.