Number-draw lotteries and keno are games whose outcome depends entirely on a random draw. The player picks a few numbers from a fixed range, the organiser draws numbers at random from the same range, and the prize depends on how many of them match. Once the ticket is filled in there are no further decisions, and the choice of numbers does not change the chance of winning. This page explains the mechanism and the calculations using two generic formats, "6 from 49" and "5 from 40", which serve here only as mathematical examples, and using keno, with 20 numbers drawn from 80.
How a number lottery works
A lottery format is described by two numbers: how many numbers are drawn and out of how many. In "6 from 49", the player marks 6 different numbers between 1 and 49 on a ticket, and the draw produces 6 numbers from the same 49. The order in which the balls come out does not matter: what counts is how many of the numbers on the ticket are among those drawn.
The draw is either mechanical, with balls mixed in a drum, or electronic, with a random number generator. In both cases the procedure is laid down in the rules of the game, and its purpose is that every number has the same chance of being drawn and that draws do not depend on one another.
Prizes are divided into tiers according to how many numbers were matched. Some lotteries pay fixed amounts for certain tiers. Others divide a prize fund among the winners of each tier, so the value of a prize also depends on how many players matched the same numbers. When nobody matches the top tier, many rules carry its fund over to the next draw. The details (the tiers, any extra numbers, the carry-over rules) differ from one lottery to another and are set by each one's rules.
How many combinations there are
A "6 from 49" ticket is a combination of 6 numbers out of 49, without order. The number of all possible combinations is given by the combinations formula:
C(49, 6) = (49 × 48 × 47 × 46 × 45 × 44) / (6 × 5 × 4 × 3 × 2 × 1) = 13,983,816
The draw produces exactly one of these combinations, and all of them have the same probability. A single ticket matches all 6 numbers with a probability of 1 in 13,983,816.
For the lower tiers, one counts the combinations that share exactly m numbers with the ticket: m numbers chosen among the 6 drawn and the rest among the 43 that were not drawn. The probability is the ratio of that count to the total number of combinations, which statisticians call the hypergeometric distribution:
P(exactly m matched) = C(6, m) × C(43, 6 − m) / C(49, 6)
| Numbers matched | Probability | About once in |
|---|---|---|
| 6 of 6 | 0.000007% | 1 in 13,983,816 |
| 5 of 6 | 0.0018% | 1 in 54,201 |
| 4 of 6 | 0.0969% | 1 in 1,032 |
| 3 of 6 | 1.77% | 1 in 57 |
| 2 of 6 | 13.24% | 1 in 8 |
The "About once in" column expresses the same probability as a ratio: over a very large number of tickets, that tier comes up on average once in that many tickets. It is not a schedule. A ticket does not get "closer" to a prize because earlier tickets did not win.
The "5 from 40" format has fewer numbers in the drum and fewer on the ticket, and therefore far fewer combinations: 658,008. A ticket matches all 5 numbers with a probability of 1 in 658,008.
| Numbers matched | Probability | About once in |
|---|---|---|
| 5 of 5 | 0.000152% | 1 in 658,008 |
| 4 of 5 | 0.0266% | 1 in 3,760 |
| 3 of 5 | 0.90% | 1 in 111 |
| 2 of 5 | 9.95% | 1 in 10 |
Comparing the two formats shows how quickly the number of combinations grows: a few more numbers in the drum or on the ticket multiply the total rather than add to it.
Every combination has the same chance
The combination 1, 2, 3, 4, 5, 6 looks almost impossible, while one such as 7, 15, 22, 31, 38, 46 looks "typical". For the drum, both are equally likely: each has a chance of 1 in 13,983,816. The impression comes from the fact that a great many combinations look "mixed" and very few look orderly. The group of mixed-looking combinations is likely precisely because it is large, but each combination in it, taken on its own, is no more likely than an orderly one.
The choice of numbers affects only one thing, and only in lotteries where a tier's fund is shared among its winners: a combination picked by many players (calendar dates, patterns drawn on the ticket, sequences) would, if drawn, lead to a prize split into more parts. The chance of matching stays the same, and so does the share of stakes paid out as prizes.
The same reasoning applies to "system" tickets, which cover every combination of 6 among a larger set of chosen numbers. Such a ticket costs as much as all the combinations it contains: its chance rises in exactly the same proportion as the amount paid, and the ratio between expected prizes and stake stays the same.
What a lottery returns
The return of a lottery, meaning the share of stakes that goes back to players in the long run, does not follow from the probabilities alone but from its rules. The organiser sets what part of the takings goes into the prize fund and how that fund is split among the tiers. The rest covers the costs of running the lottery and the other purposes laid down in the rules or by law. The share set aside for prizes is published by the organiser, usually in the rules of the game.
The probabilities in the tables show how often each prize tier comes up; the average value of a ticket depends on how much each tier pays. In the long run, players as a whole get back the share of stakes set aside for prizes, and the difference is the organiser's edge, the counterpart of the house edge in casino games. For any single player, results are extremely spread out: a great many lose their stake, very few win large amounts. The general mechanism is explained in House edge and RTP.
Keno
Keno uses 80 numbers, of which 20 are drawn, with a drum or a random number generator. The player chooses how many numbers to mark on the ticket, usually between 1 and 10, depending on the rules of the game. The paytable differs according to how many numbers were picked: a 4-number ticket has its own prizes for 2, 3 or 4 matches, a 10-number ticket has others.
Each number picked has a chance of 25.00% of being among those drawn. The probability that all the numbers picked are drawn, however, falls very quickly as the ticket holds more numbers:
| Numbers picked | All matched | None matched |
|---|---|---|
| 1 | 1 in 4 | 75.00% |
| 2 | 1 in 17 | 56.01% |
| 3 | 1 in 72 | 41.65% |
| 4 | 1 in 326 | 30.83% |
| 5 | 1 in 1,551 | 22.72% |
| 6 | 1 in 7,753 | 16.66% |
| 8 | 1 in 230,115 | 8.83% |
| 10 | 1 in 8,911,711 | 4.58% |
The formula is the same as for the lottery. The probability of matching exactly m of k numbers picked is:
P(exactly m matched) = C(k, m) × C(80 − k, 20 − m) / C(80, 20)
The "None matched" column shows a peculiarity of the game: on tickets with many numbers, matching none becomes relatively rare, which is why some paytables award a prize for zero matches too. Rules of this kind differ from one game to another.
The return of keno depends entirely on the paytable, which differs from one game to another and even between tickets with different counts of numbers in the same game. The rules published by the organiser state it. Keno draws often take place at short intervals. The pace does not change the organiser's edge on each ticket, but more draws per hour mean more stakes exposed to the same edge and, therefore, a larger average loss over the same period of time.
The myth of "hot" and "cold" numbers
Statistics of past draws always show numbers that came up more often ("hot") and numbers that came up less often ("cold"). The differences are normal: over a finite number of draws, frequencies are never perfectly equal, just as ten tosses of a coin rarely give exactly half heads and half tails.
The drum has no memory. A ball does not "know" how many times it has been drawn and is not "owed" any appearance. If the draws are fair, every number has the same chance at every draw, whatever the history. In the long run the relative frequencies of the numbers move closer together, not because the "cold" numbers catch up, but because a very large number of new draws dilutes the old deviations. The confusion has a name, the gambler's fallacy, and its link with large numbers of repetitions is explained in Variance and the law of large numbers.
A number that came up unusually often over a very large number of draws could, in theory, point to a fault in the equipment; this is why draws are usually carried out under verification procedures. The history of the draws, however, holds no information about the next draw.