Raffle numbers

The word reaches English as Middle English rafle, from Old French rafle, which meant both a dice game and plunder, and behind it the verb rafler, to snatch or carry off. The obsolete English sense names a specific game: three dice, and whoever throws three of a kind takes the stakes. The ticket, though, comes from the lottery rather than the dice cup. The earliest recorded traces of a lottery are keno slips from the Chinese Han dynasty, dated between 205 and 187 BC.

Lotteries with cash prizes are a fifteenth-century Low Countries invention, run by towns to pay for walls and poor relief. A record dated 9 May 1445 at L'Ecluse describes a draw of 4,304 tickets with 1,737 florins in prize money, raised to build town fortifications. England's first official lottery was chartered by Elizabeth I in 1566 and drawn in 1569; a ticket cost ten shillings, roughly three weeks' wages for an ordinary worker, against a top prize of about £5,000. At the start of the Revolutionary War the Continental Congress turned to lotteries to fund the Colonial Army.

The barrel supplied the raffle's other British name. Tombola is a lottery-style board game from southern Italy, now mostly a Christmas and New Year fixture and usually dated to 1734, when the new king of Naples began taxing winnings on the Lotto; English borrowed the word for the tumbling drum, so that in British usage a tombola is simply a raffle whose tickets are churned in a barrel before the draw.

What a barrel is for is a fair shuffle. The method used to simulate one was published by Ronald Fisher and Frank Yates in 1938 in Statistical Tables for Biological, Agricultural and Medical Research, using a printed table of random numbers; Richard Durstenfeld recast it as an in-place linear-time algorithm in Communications of the ACM in 1964, and Donald Knuth popularised that version in The Art of Computer Programming as Algorithm P.

Key properties

• Tickets are drawn without replacement: each winning ticket leaves the barrel, so no ticket can take two prizes.
• In a simple random sample of k tickets from n, every ticket has probability k/n of being drawn, and every set of k tickets is equally likely.
• Holding m of n tickets gives a 1 − C(n−m, k)/C(n, k) chance of winning at least one of k prizes. That is at most m·k/n, and strictly below it whenever m > 1 and k > 1.
• With a single prize the odds are exactly m/n, so doubling your tickets exactly doubles your chance; with several prizes it falls slightly short of doubling.
• Order matters once the prizes differ: drawing k tickets from n has n!/(n−k)! ordered outcomes but only C(n, k) unordered ones.
• A 32-bit seeded generator reaches at most 2³² ≈ 4.29 billion distinct draw sequences. A 1,000-ticket raffle already exceeds that at four winners, since 1000 × 999 × 998 × 997 is about 9.9 × 10¹¹.
• In Great Britain a raffle is a lottery in law. Raffles held at an event can qualify as incidental lotteries under the Gambling Act 2005, which require physical tickets sold at the event and cap expenses taken from the proceeds at £100.

Where they turn up

• School fetes, church fairs and charity dinners run the classic paper-ticket raffle, which in Britain is the "incidental lottery" the Gambling Commission describes — no licence needed if the conditions are met.
• "Tombola" is the usual British name for the same draw when the tickets are tumbled in a barrel first.
• Conference and trade-show prize draws use the identical mechanism with a badge-scan list standing in for the barrel.
• Oversubscribed ballots — marathon places, festival tickets, some school admissions — are structurally the same draw: winners taken without replacement from a pool of applicants.
• Cash raffles are illegal in some jurisdictions, California among them with limited exceptions, which is one reason so many prize draws offer goods rather than money. Check local rules rather than relying on this page.