Keno odds: picks, hits and the hypergeometric distribution
Exact keno odds for 1 to 10 picks from 40 numbers, why 10 of 10 is far rarer than in 80-ball keno, and what paytables and risk levels change.
Fact-checked 2 November 2026 · 8 sourcesKeno works like a small lottery: you pick some numbers, the game draws its own, and a paytable pays according to how many of your picks were drawn. With 40 numbers and 10 drawn, every chance can be counted exactly, and the counts show what a paytable is worth and what a risk level changes.
How keno works
Casino keno is usually played with the numbers 1 to 80, and 20 are drawn at random. Each of your numbers that comes up is a hit, or a catch in keno terms, and the payout depends on how many numbers you picked and how many you caught. Each casino sets its own paytables, the lists of what every hit count pays, and they vary widely.
The game is of Chinese origin: early tickets carried the first 80 characters of the Thousand Character Classic instead of numbers, and Chinese immigrants brought it to the United States in the 19th century. The name comes from French or Latin words for five, quine and quini.
Some online keno games shrink the board to 40 numbers and draw 10. cs2.casino uses this version: you pick 1 to 10 numbers from 1 to 40, 10 different numbers are drawn, and you choose a risk level, Classic, Low, Medium or High. Both formats draw a quarter of the board, so any single pick comes up 25% of the time. Which numbers you pick makes no difference: every number is equally likely, so only how many you pick matters.
Counting hits
To find the chance of h hits with p picks, count draws. Choose which h of your numbers are among the 10 drawn, fill the other 10 − h places from the 40 − p numbers you didn't pick, and divide by all possible draws:
P(h hits) = C(p, h) × C(40 − p, 10 − h) / C(40, 10)
C(n, k) is the number of ways to choose k items from n, and there are C(40, 10) = 847,660,528 possible draws. This is the hypergeometric distribution, the math of drawing without putting anything back; 80-ball keno uses the same formula with 80 and 20.
Take 5 picks and 2 hits. There are C(5, 2) = 10 ways to choose which two of your numbers come up, and C(35, 8) = 23,535,820 ways to fill the other 8 places from the 35 you didn't pick. That's 235,358,200 draws out of 847,660,528, a 27.77% chance.
On average a quarter of your picks are drawn, so 10 picks catch 2.5. Mines game odds applies the same distribution to the case where every pick has to be safe.
Hit chances for 1 to 10 picks
Exact chances, rounded:
| Picks | No hits | Likeliest result | All picks hit | |--:|--:|--:|--:| | 1 | 75.00% | 0 hits, 75.00% | 1 in 4 | | 2 | 55.77% | 0 hits, 55.77% | 1 in 17.3 | | 3 | 41.09% | 1 hit, 44.03% | 1 in 82.3 | | 4 | 29.99% | 1 hit, 44.42% | 1 in 435 | | 5 | 21.66% | 1 hit, 41.65% | 1 in 2,611 | | 6 | 15.47% | 1 hit, 37.13% | 1 in 18,278 | | 7 | 10.92% | 2 hits, 34.40% | 1 in 155,363 | | 8 | 7.61% | 2 hits, 34.74% | 1 in 1,708,993 | | 9 | 5.23% | 2 hits, 33.50% | 1 in 27,343,888 | | 10 | 3.54% | 2 hits, 31.07% | 1 in 847,660,528 |
Up to two picks, missing everything is the likeliest result; from three to six picks it's one hit, and from seven to ten it's two. For 10 picks the whole spread looks like this:
| Hits | Draws out of 847,660,528 | Chance | |--:|--:|--:| | 0 | 30,045,015 | 3.54% | | 1 | 143,071,500 | 16.88% | | 2 | 263,381,625 | 31.07% | | 3 | 244,296,000 | 28.82% | | 4 | 124,692,750 | 14.71% | | 5 | 35,911,512 | 4.24% | | 6 | 5,755,050 | 0.68% | | 7 | 487,200 | 0.057% | | 8 | 19,575 | 0.0023% | | 9 | 300 | 1 in 2.8 million | | 10 | 1 | 1 in 847.7 million |
Five or more hits come up in 4.98% of draws, about 1 in 20.
Why 10 of 10 is so rare with 40 numbers
Both formats draw a quarter of the board, and 10 picks average 2.5 hits in each. The extremes differ. In 80-ball keno a 10-spot ticket, one with 10 picks, catches all 10 once in 8,911,711 draws; with 40 numbers it's once in 847,660,528, about 95 times rarer.
The reason is room in the draw. With 20 numbers drawn, a perfect 10-spot still leaves 10 drawn numbers free. With 10 drawn, the draw must be exactly your ten numbers. The 40-number game is tighter at both ends: zero hits from 10 picks is rarer too, 3.54% against 4.58%, and the standard deviation, the typical distance from the average, is 1.20 hits against 1.29.
The gap shrinks with fewer picks: 5 of 5 is 1 in 2,611 with 40 numbers and 1 in 1,551 with 80, and a single pick is 1 in 4 in both.
From paytable to return
A paytable's return to player, or RTP, is each multiplier times the chance of its hit count, added up, and the house edge is 1 − RTP. House edge and RTP covers the idea; keno just has one term per hit count. With one pick a hit is 25%, so a fair table pays 4× for a hit. At a 1% edge, as an example, it pays 0.99 / 0.25 = 3.96×.
Live keno has long been one of the costliest bets in a casino. Wikipedia works through a typical $1 6-spot ticket in 80-ball keno that returns the dollar for 3 hits and pays $4 for 4, $88 for 5 and about $1,500 for 6: weighted by their chances, those pays return about 71%, a house edge of 29%. The Wizard of Odds' 2001 survey of the 9-spot ticket at every live keno game in Las Vegas found returns of 65% to 80%, and its 2008 survey of video keno in San Diego found 84% to 95%.
Each pick count has its own table, so the return can also change with the number of picks: the Wizard of Odds' analysis of one 40-number online game puts its return at 95.66% to 97.90%, depending on the pick count. cs2.casino shows each game's house edge and bet limits, and over time the house wins whichever table you play.
What a risk level changes
A risk setting swaps the paytable, not the draw: the numbers drawn and every chance above stay the same. Here are two example tables for 2 picks at a 1% edge, worked out for this article. Two picks catch nothing 55.77% of the time, one number 38.46% and both 5.77%.
| 2 picks | 0 hits | 1 hit | 2 hits | Return | |---|--:|--:|--:|--:| | Table A | 0 | 0 | 17.16× | 99% | | Table B | 0 | 1.5× | 7.16× | 99% |
Table A puts everything on the rare double: 17.16 × 5.77% = 99%. Table B also pays for one hit: 1.5 × 38.46% + 7.16 × 5.77% = 57.69% + 41.31% = 99%. Both keep 1% of every coin wagered on average. A pays in one round of 17.3 and B in 44% of rounds, and over 100 one-coin rounds the typical swing (one standard deviation) is about ±40 coins for A and ±17 for B.
A riskier table means bigger, rarer wins and longer dry spells, not a different average.
Hot numbers and checking a draw
A number that came up in recent rounds is no more likely in the next one, and one that hasn't isn't due. On cs2.casino, Keno is a solo game with a seed pair: a server seed committed by its SHA-256 hash before you bet, a client seed you can change, and a nonce that goes up by one each bet. Past draws aren't an input; the gambler's fallacy explains why streaks feel otherwise.
The floats come from HMAC-SHA256(server seed, "client seed:nonce:k") and pick the 10 numbers one at a time from those still left, the partial draw the Fisher–Yates shuffle article describes. Rotate your seed pair and the old server seed is revealed, so every draw made with it can be checked in the provably fair calculator. The game is at Keno.
