Keno odds and paytables

The draw: 10 from 40

Lumenbet Keno uses a 40-number grid. You pick between 1 and 10 numbers, then the game draws 10 numbers without replacement. Your result is simply the number of hits — picks that appear among the 10 drawn — and the paytable for your pick count turns that hit count into a payout.

There are C(40,10) = 847,660,528 possible draws, all equally likely. Everything else is counting.

The hypergeometric formula

This is sampling without replacement, so the relevant distribution is hypergeometric rather than binomial. If you pick k numbers and the game draws 10 from 40, the chance of exactly h hits is:

P(h hits) = C(10, h) × C(30, k − h) / C(40, k)

Read it as: choose which h of your picks land among the 10 drawn numbers, choose the remaining k − h picks from the 30 numbers that were not drawn, and divide by every way of choosing k numbers from 40.

A useful shortcut: 10 of the 40 numbers are drawn, so each individual pick has a 25% chance of being a hit, and your expected hit count is k/4. Pick 4 numbers and you average one hit. Pick 10 and you average 2.5. That average never changes, which is the first clue that picking more numbers cannot buy you an edge.

Pick 10: the full distribution

The 10-pick grid is where the big multipliers live, so it is worth seeing the whole shape of it.

Hits	Combinations	Probability	Roughly
0	30,045,015	3.545%	1 in 28
1	143,071,500	16.88%	1 in 6
2	263,381,625	31.07%	1 in 3.2
3	244,296,000	28.82%	1 in 3.5
4	124,692,750	14.71%	1 in 6.8
5	35,911,512	4.237%	1 in 24
6	5,755,050	0.679%	1 in 147
7	487,200	0.0575%	1 in 1,740
8	19,575	0.00231%	1 in 43,303
9	300	0.0000354%	1 in 2,825,535
10	1	0.000000118%	1 in 847,660,528

Three quarters of all 10-pick rounds land on 1, 2 or 3 hits. Seven hits or better happens about once in 1,700 rounds. That is why a 10-pick table can afford a huge top multiplier and still return the same 96% as every other pick count.

Hitting everything you picked

The headline odds for each pick count are C(10,k) / C(40,k) — the chance that every number you chose is drawn.

Picks	All hits	Probability
1	1 in 4	25%
2	1 in 17.3	5.77%
3	1 in 82.3	1.21%
4	1 in 435	0.230%
5	1 in 2,611	0.0383%
6	1 in 18,278	0.00547%
7	1 in 155,363	0.000644%
8	1 in 1,708,993	0.0000585%
9	1 in 27,343,888	0.00000366%
10	1 in 847,660,528	0.000000118%

Each extra pick divides the perfect-ticket chance by roughly four to thirty. That steep fall is what the top multipliers are paying for.

How a paytable is solved to a 4% edge

Keno on Lumenbet has a 4% house edge, so every paytable is built so that the sum of probability times payout equals 0.96:

expected return = Σ_h P(h hits) × payout(h) = 0.96

The probabilities are fixed by the maths above, so the designer only chooses how to spread 96% of the stake across the hit counts. Here are two made-up solutions, not Lumenbet's multipliers — the real ones are printed on the game screen, but the arithmetic is identical. A payout of 2× means two units returned for one staked, so 1× is break-even.

Pick 2. P(1 hit) = 38.46%, P(2 hits) = 5.77%. Pay 1× on one hit and 9.97× on two:

0.3846 × 1 + 0.0577 × 9.97 = 0.3846 + 0.5752 = 0.9598 ≈ 0.96

Pick 3. P(2 hits) = 13.66%, P(3 hits) = 1.215%, nothing paid below two hits. Pay 2.5× and 50.9×:

0.1366 × 2.5 + 0.01215 × 50.9 = 0.3416 + 0.6184 ≈ 0.96

Notice what happened in the second example: removing the small payout for one hit freed up value, and all of it went into the rare outcomes. That is the only lever a keno paytable has. Some keno games elsewhere even pay something for zero hits on large pick counts (Lumenbet's table does not), because missing everything with 9 or 10 picks is itself unlikely; that payout is funded by trimming the middle of the table.

Does picking more numbers improve your odds?

No. Every pick count on Lumenbet returns 96% (a 4% house edge), so the choice is purely a variance decision:

• Few picks (1–3): frequent small results, shallow swings, no large multipliers.
• Many picks (8–10): most rounds pay nothing or very little, with the whole value concentrated in results that arrive once in thousands of rounds.

With one pick you win 25% of the time. With ten picks you miss everything 3.5% of the time and land fewer than four hits in about 80% of rounds. Same expected cost, very different ride. If you want the full picture of what that does to a balance over a session, see bankroll and volatility.

And the arithmetic does not care how you choose the numbers. Consecutive numbers, birthdays, last round's draw, a quick pick — every set of k numbers has exactly the same distribution of hits, because every draw is equally likely. Numbers are not "due". Doubling stakes after a dry run does not change the 4%; it only changes how fast you reach a number you did not want to see. If that pattern sounds familiar, read responsible gambling.

The provably fair draw

The 10 drawn numbers are not generated at random on the server when you press play. They are derived from HMAC-SHA256(server seed, "client seed:nonce"), where the SHA-256 hash of the server seed was published before you bet and the seed itself is revealed when you rotate it.

Turning a hash into 10 distinct numbers from 40 needs care. Simply taking a byte modulo 40 would make some numbers slightly more likely than others, because 256 is not divisible by 40. Keno instead uses rejection sampling: values that fall outside a clean range are discarded rather than folded back in, and duplicates are skipped until 10 distinct numbers are collected. The result is a uniform draw with no modulo bias, and because the whole process is deterministic you can recompute any past round yourself.

The method, the verifier and worked examples are at the provably fair page, with the background in our explainer.

What to hold on to

Hit probabilities come from one hypergeometric formula. Expected hits are always a quarter of your picks. Every paytable, for every pick count, returns 96%, which means the 4% edge is unavoidable and compounds over volume — in the long run the house edge wins, and no pick pattern changes that. The only thing you are genuinely choosing is the shape of your variance.