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Lotteries are older than the mathematics used to analyse them. Records from the Low Countries describe money-prize draws in the 1440s; a notice from L'Ecluse, now Sluis, dated 1445 mentions 4,304 tickets sold to pay for the town's fortifications. Elizabeth I chartered England's first state lottery in 1566, and it was drawn in 1569 at the west door of St Paul's Cathedral, the proceeds earmarked for harbours and other public works.

The shape most modern games still use traces to Genoa. The standard account is that Genoese bettors wagered on which five of ninety candidates would be drawn to fill seats in the city's governing council, and that when the betting outgrew the elections it was run with numbered balls instead. Whether the story is exact is hard to establish, but the ninety-ball drum itself survives in Italy's Lotto and in SuperEnalotto, and the same idea produced keno and bingo. The probability theory arrived much later: the 1654 correspondence between Blaise Pascal and Pierre de Fermat about how to divide the stakes in an interrupted game is usually taken as the start of the subject, which means people had been buying tickets for two centuries before anyone could calculate what one was worth.

Letting the player choose the numbers is recent. Canada's Lotto 6/49, launched in June 1982, was the first nationwide Canadian game in which players picked their own six numbers instead of buying a pre-printed ticket, and the design spread fast: the UK National Lottery opened on 19 November 1994 with a 6/49 draw, widened to 6/59 in October 2015.

Choice invited arithmetic. Over the May bank holiday weekend of 1992 a syndicate organised by the Dublin accountant Stefan Klincewicz set out to buy a large share of the 1,947,792 combinations of Ireland's then 6-from-36 Lotto, having worked out that the rolled-over jackpot plus the lower prize tiers came to more than the tickets would cost. The operator's answer was structural: before the end of August 1992 the Irish game was widened to 6 from 39 and a bonus ball was added, and the matrix has been widened three more times since, reaching 6 from 47 in September 2015. Where draws have actually gone wrong, it has been through tampering rather than mathematics: the Pennsylvania Lottery's "Triple Six Fix" of 24 April 1980 used weighted balls to force 666 in the daily number.

主な性質

• Every allowed combination is exactly as likely as every other one. No picker, system, seed or set of "lucky" numbers changes the probability that a line wins — this page only chooses which equally-likely line you hold.
• Powerball has C(69,5) = 11,238,513 main combinations, and multiplying by the 26 Powerballs gives 292,201,338 possible tickets.
• Mega Millions in its April 2025 matrix has C(70,5) = 12,103,014 main combinations and 24 Mega Balls, so 290,472,336 tickets.
• EuroMillions has C(50,5) = 2,118,760 main combinations and C(12,2) = 66 Lucky Star pairs, giving 139,838,160 tickets.
• UK Lotto has C(59,6) = 45,057,474 lines; Canada Lotto 6/49 has C(49,6) = 13,983,816; Italy SuperEnalotto has C(90,6) = 622,614,630.
• Main numbers are drawn without replacement, so no line ever repeats a number. A second-pool number such as the Powerball, Mega Ball or SuperStar is drawn from its own pool and can therefore equal one of the main numbers.
• Bonus balls — the UK bonus ball, the Canadian bonus, Australian supplementaries, Italy’s Jolly, Spain’s complementario — are taken from what is left of the main drum after the main draw, so they are never marked on a slip. This page shows them only in simulated-draw mode.
• Output here comes from a seeded mulberry32 generator, not from a hardware source: the same seed and format always reproduce the same lines, and it is not cryptographically secure.

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• In the Powerball draw of 30 March 2005, 110 tickets matched all five white balls. The combination had been printed on slips inside fortune cookies from a single manufacturer, and the sixth number on the slip did not match that night’s Powerball. Second prize was a fixed amount rather than a shared pool, so all 110 were paid in full — 89 at $100,000 and 21 at $500,000 under the Power Play option — and the lottery paid out far more at that tier than a normal draw.
• Ticket data repeatedly shows players over-selecting numbers from 1 to 31 and avoiding high numbers, because so many people play birthdays. This is observed player behaviour, not a flaw in the draw: it leaves the chance of winning unchanged and only raises the expected number of people you would share a jackpot with.
• "Hot" and "cold" number charts, wheeling systems, dream books and numerological picks are a widespread belief about lotteries rather than an established effect. In a fair draw the balls have no memory, so past results carry no information about the next draw.
• Terminal-generated numbers are a standard product feature: Quick Pick in the United States and Canada, Lucky Dip in the United Kingdom, machine selection elsewhere. They exist because choosing numbers by hand is the slow part of selling a ticket.
• The 6-from-49 draw is the stock worked example for combinations in school and undergraduate probability courses, which is why C(49,6) = 13,983,816 turns up in textbooks far more often than in anyone’s actual winnings.
• Most countries run lotteries as a state or licensed monopoly with part of the stake money earmarked for public spending — good causes in the United Kingdom, education budgets in several US states — which is why the odds and prize structures are set out in published regulations.