Keno Odds Calculator

Leave Numbers Picked at 5 and Matches at 3. Calculate. Match Probability is 8.3935% and Odds (1 in N) is 1:12. That is the hypergeometric probability of catching exactly three of your five spots when the house draws 20 balls from 80. Combinations: C(20,3) × C(60,2) / C(80,5). Numerically, 1,140 × 1,770 / 24,040,016 ≈ 0.083935, which the tool prints as 8.3935% and as 1-in-12 after rounding 1/p to the nearest integer.

Change Matches to 5 to price a five-spot catch-all. Probability collapses to 0.0645%, odds 1:1551. That is the ticket people mean when they say they "hit their keno." Change Matches to 0: 22.7184%, 1:4 — missing everything on a five-spot is common, not a scandal. Matches 1: 40.5686%, 1:2. Matches 2: 27.0457%, 1:4. Matches 4: 1.2092%, 1:83. Those five rows are the full distribution of a five-spot ticket; they add to 100% aside from rounding.

Numbers Picked runs from 1 to 10, the usual casino ticket sizes. A ten-spot with Matches 10 is a different hypergeometric: C(20,10)×C(60,0)/C(80,10), a much smaller probability than the five-spot jackpot. Matches is clamped to at most Picks, so asking for 8 matches on a 5-spot ticket is treated as 5. The draw is always 20 out of 80; this is not a 10-ball quick-keno variant and not a 80-pick live draw with a different catch.

The calculator reports the probability of exactly k matches, not "k or more." If a paytable awards a prize for 3, 4, and 5 on a five-spot, you must run the tool three times and weight each probability by that prize to get EV. A typical five-spot that pays $2 on a $1 ticket for three spots, $20 for four, and $400 for five is a negative-EV product even though 8.3935% feels like "almost one in ten."

Do not confuse this with lottery hypergeometric tools that draw 6 from 49. Keno's 20-from-80 catch is denser: you are supposed to hit some of your spots. The house edge lives in the paytable, not in an unusually cruel draw. Use the 1-in-N figure to read a glass prize, then compute EV separately.

Keno is a 20-ball draw from an 80-number field. The player marks 1 to 15 spots (this tool caps at 10) on a ticket; the house draws twenty; prizes depend on how many of the marked spots appear in the draw. The combinatorial model is hypergeometric, the same urn model as a lottery, with much larger "drawn" and "picked" parameters. That density is why a five-spot ticket hits exactly three numbers about 8.4% of the time and hits nothing about 22.7% of the time — both of which surprise people who imported their intuition from 6/49.

The game's paper trail runs through Chinese lotteries of the Qing era, the 19th-century "white pigeon" tickets, and the American keno rooms that sat next to bingo halls in Nevada. Western casino keno kept the 80-spot card and the 20-spot draw and then printed paytables with house edges of 25 to 40 percent, among the worst main-floor products still dealt. Way tickets, king tickets, and multi-game books do not change the underlying hypergeometric draw; they only bundle several hypergeometric random variables onto one piece of paper.

Live keno, video keno, and "mini keno" that draws 10 or 12 balls are different games. A 10-ball draw from 80 makes every catch-all far less likely and usually comes with a different paytable. This calculator is the 20-from-80 standard. If your app draws 10, the numbers here are not yours. If your state lottery uses 80/20 branding on a daily draw, they are.

The calculator does not ingest a paytable. Two casinos can post identical 8.3935% three-spot frequencies and pay $1 versus $3 on a $1 five-spot three-catch, which is the difference between a 30% hold and a 10% hold. Probability is the easy half. Pricing is the half that actually costs money. Ancient Chinese tickets and modern Nevada racebooks share the urn; they do not share the prize.

Limits: independent games, without replacement inside a game, with replacement across games (a new 80-ball field each draw). No catch-all parlay across consecutive games, no way-ticket covariance, no multi-race "20-spot" specials. Exact-k only. For "at least 3" on a five-spot, add the k=3,4,5 rows: 8.3935 + 1.2092 + 0.0645 ≈ 9.667%.

Casino keno is a hypergeometric experiment, not a sequence of independent coin flips. Eighty numbered balls form the field. The house draws twenty of them. The player has already marked n spots on a ticket, and the question is the chance that exactly k of those marked spots appear among the twenty. Sampling is without replacement: once a ball is drawn it cannot return, so the usual binomial shortcut is the wrong model. Pascal and Fermat’s combinatorial counting, later written as binomial coefficients, is the right one.

The probability of exactly k matches is the product of two ways of filling the ticket, divided by the ways of choosing any n marks from eighty. Choose k of the twenty drawn balls to land on the marked spots, then choose the remaining n minus k marks from the sixty balls that stayed in the hopper. Divide by the number of n-subsets of eighty. After that sentence the identity is C(20, k) times C(60, n − k) over C(80, n). The legal range for k is from the larger of zero and n minus sixty up to the smaller of n and twenty. Outside that range the probability is exactly zero.

Work the published default rather than a ten-spot rumour. Numbers picked equal five and matches equal three. The three coefficients are C(20, 3) = 1,140, C(60, 2) = 1,770, and C(80, 5) = 24,040,016. The numerator is 1,140 × 1,770 = 2,017,800. Division yields 2,017,800 / 24,040,016 ≈ 0.083935, reported as a match probability of 8.3935 percent and as odds of 1 in 12 after the reciprocal 11.91 is rounded to the nearest integer. A catch-all five-spot, k = 5, collapses to C(20, 5) / C(80, 5) ≈ 0.0645 percent, or 1 in 1,551. A ten-spot that catches all ten is a different, much thinner cell of the same family; the worksheet default is the five-and-three cell.

The rest of the five-spot distribution places that 8.3935 percent in context. Catching nothing is about 22.7184 percent (roughly 1 in 4). Catching one is 40.5686 percent (about 1 in 2). Catching two is 27.0457 percent (about 1 in 4). Catching four is 1.2092 percent (1 in 83). The six masses add to one aside from display rounding. Players who imported their intuition from a six-from-forty-nine lottery are routinely surprised that missing everything on a five-spot is a common event, not a scandal.

Assumptions are strict and should be stated. The draw is a uniform random twenty-subset of an eighty-set. Way tickets and king tickets are unions of these cells, not a different urn. Multi-race books are products of independent draws if the balls are re-mixed. The calculation returns only the probability of a match count; it does not price a Nevada paytable whose house share often sits between 25 and 40 percent. Coefficients are evaluated as falling factorials over k-factorial so they remain inside ordinary floating-point range for n at most ten. History of the matrix runs through Chinese lotteries, nineteenth-century pigeon tickets, and American keno rooms beside bingo halls. None of that folklore changes the ratio of binomial coefficients.