The Draw That Defines the Odds
Every keno ticket starts with the same structural fact: the house draws 20 numbers from a pool of 80. You mark anywhere from 1 to 20 spots on your card, and your payout depends on how many of your marked numbers appear among the 20 drawn. The odds of any particular outcome follow directly from that fixed structure. Nothing about the game's presentation, the speed of the draw, or the size of the prize changes the underlying probability of a match. What changes is how the operator prices the winning combinations, and that is where the house edge lives.
Computing keno odds is a counting problem. You count the number of ways a specific match can happen, then divide it by the total number of ways 20 balls can be drawn from 80. The branch of mathematics that handles this is combinatorics, and the specific distribution is called the hypergeometric distribution. It applies whenever you sample without replacement from a finite population that contains a known number of 'successes'—in this case, your marked spots sitting among the 80 possible numbers.
Setting Up the Counting Problem
Suppose you mark n spots on your ticket. The draw selects 20 numbers from the 80 available. You want to know the probability that exactly k of your n spots appear in the draw. The total number of possible draws is the number of ways to choose 20 items from 80, written as C(80, 20). This is a fixed denominator for every keno calculation, regardless of how many spots you play. It does not change from game to game or operator to operator.
The numerator counts the draws that produce exactly k matches. To build such a draw, the 20 selected numbers must include k of your n marked numbers and 20 − k numbers from the 80 − n numbers you did not mark. The number of ways to choose k numbers from your n spots is C(n, k). The number of ways to choose the remaining 20 − k numbers from the unmarked pool is C(80 − n, 20 − k). Multiply those two counts together, and you have the number of draws that give exactly k matches.
The probability of exactly k matches is therefore:
C(n, k) × C(80 − n, 20 − k) ÷ C(80, 20)
Each C(a, b) term is a binomial coefficient, calculated as a! ÷ (b! × (a − b)!), where the exclamation mark denotes a factorial. The arithmetic becomes large quickly—C(80, 20) is a number with many digits—but the formula itself is compact and exact.
Worked Example: Matching 4 of 6 Spots
As an example, suppose you mark 6 spots and want the probability of matching exactly 4 of them. The denominator is C(80, 20). The numerator is C(6, 4) × C(74, 16), because you must choose 4 of your 6 marked numbers and 16 of the 74 numbers you did not mark. C(6, 4) equals 15. C(74, 16) is the number of ways to pick 16 numbers from 74. Dividing the product by C(80, 20) gives the probability, which works out to roughly 0.00287, or about 1 in 348. That is the chance of exactly four matches on a six-spot ticket.
If you instead want the probability of matching at least 4 of your 6 spots, you would sum the probabilities for exactly 4, exactly 5, and exactly 6 matches. The 'at least' version is what most payout tables care about, because a ticket that matches 5 of 6 typically pays a larger prize than one matching 4 of 6, and the operator prices each tier separately.
How the Number of Spots Changes the Distribution
Keno odds shift dramatically as you change the number of spots you mark. The comparison below shows what changes and what stays fixed across the three most common ticket sizes. The denominator of every calculation is identical; only the numerator terms adjust.
| Ticket size | What changes in the formula | What stays fixed | Practical effect on play |
|---|---|---|---|
| 1 spot | Only C(1, k) and C(79, 20 − k) appear in the numerator | Denominator C(80, 20); draw size 20 | Simplest odds; a match is either 0 or 1, so probability is just 20/80 |
| 6 spots | Numerator uses C(6, k) and C(74, 20 − k) | Denominator C(80, 20); draw size 20 | Most likely outcome is 1 or 2 matches; higher match counts become rare |
| 20 spots | Numerator uses C(20, k) and C(60, 20 − k) | Denominator C(80, 20); draw size 20 | Matching all 20 is astronomically rare; many lower-tier matches are common |
The most probable number of matches on any ticket is not simply proportional to the spots marked. For a six-spot ticket, the single most likely outcome is one match, not three. For a twenty-spot ticket, the most likely outcome is five matches, because the draw contains 20 of the 80 numbers, and 20 is one quarter of 80. The distribution clusters around that expected value, but the tails—especially the high-match tail—fall away extremely fast.
Why the Payout Table Is a Separate Calculation
Computing the odds of a match is only half the picture. The other half is how much the operator pays for each match tier, and that is where the game's return rate is determined. A keno game could pay 3-for-1 on a four-of-six match, or 4-for-1, or nothing at all for lower tiers. The probability of the match is fixed by the draw structure; the payout is set by the operator. The house edge on a keno ticket is the difference between the true odds of an outcome and the odds the payout table offers.
To compute the expected return of a specific ticket, you multiply each match tier's probability by its payout, then sum across all tiers. If the sum is below the wager, the difference is the house edge. Different operators offer different payout tables for the same spot count, which is why two keno games with identical draw mechanics can have materially different long-run returns. The game information panel or rules page for the specific keno product states the payout table, and that is the document to read before judging any ticket.
What the Formula Cannot Tell You
The hypergeometric calculation gives you exact probabilities for any spot count and any match tier. It does not give you a winning strategy, because no selection of numbers changes the probability of a match. Every number from 1 to 80 has the same chance of being drawn, and the draw is independent from game to game. A number that has appeared frequently in recent draws is not more or less likely to appear in the next one. The odds of matching k spots depend only on how many spots you marked and how many the house draws, never on which specific numbers you chose.
Keno is a negative-expectation game in essentially every commercial setting. The payout table is built so that the average return per wager is below 100% of the wager, and no betting pattern or number-selection method changes that. The value of computing the odds is clarity: you can see exactly how rare a high-match outcome is, and you can compare payout tables across products to find the one that returns the largest share of wagers over the long run. That is a choice about the rate of loss, not a route to profit.
FAQ
What is the probability of matching all 20 numbers in keno?
The probability is 1 divided by C(80, 20), because there is exactly one draw that contains all 20 of your marked numbers. That is an extraordinarily small number, far beyond any realistic expectation of occurrence in a lifetime of play. It is the same probability regardless of which 20 numbers you mark.
Does the keno odds formula change if the game draws fewer or more than 20 numbers?
Yes. The formula generalises to any draw size: if the house draws d numbers from a pool of p, and you mark n spots, the probability of exactly k matches is C(n, k) × C(p − n, d − k) ÷ C(p, d). The standard game uses p = 80 and d = 20, but the structure works for any variation. Always check the specific game's rules for its pool size and draw size before applying the formula.
Can I compute keno odds without doing the factorial arithmetic by hand?
Yes. Most spreadsheet programs include a hypergeometric distribution function, often named HYPGEOM.DIST or similar, that accepts the population size, the number of successes in the population, the sample size, and the number of observed successes. You can also compute binomial coefficients with a scientific calculator or a programming language's combinatorics library. The key is to keep the four inputs straight: 80 total numbers, your marked spots as the successes, 20 drawn, and the match count you are testing.
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