Keno Odds Explained: How Many Numbers to Pick

Keno is a lottery-style game. You pick between 1 and 10 numbers from a pool, the game draws a set of numbers, and you are paid according to how many of your picks were drawn. In Cashout Arcade the pool is the numbers 1 to 40, the game draws 10, and you pick 1 to 10. The odds follow from simple counting, the pay tables are built so that every combination of picks and risk returns the same 97%, and no number is any luckier than another.

How a keno round works

1. You pick 1 to 10 numbers from 1 to 40, and choose a risk level (low, medium or high).
2. The server draws 10 different numbers from the 40, with every number equally likely.
3. Your hits are the picks that were among the 10 drawn.
4. The pay table for your number of picks and risk level gives the multiplier for that number of hits.

Each draw is made with a secure random generator on the server (see the RNG guide), and nothing about your chosen numbers influences it.

The probability of each number of hits

The chance of hitting exactly h of your k picks follows the hypergeometric distribution:

P(h hits) = C(10, h) x C(30, k - h) / C(40, k)

where C(n, r) is the number of ways to choose r items from n. Worked example, 5 picks and exactly 3 hits: C(10, 3) = 120 ways to choose which three of the ten drawn numbers you hit, C(30, 2) = 435 ways for the other two picks to miss, and C(40, 5) = 658,008 ways to choose your five picks. The chance is 120 x 435 / 658,008 = 7.933%.

Here is the full table for every number of picks (exact, from the same formula the game uses):

Two things the table makes clear. The expected number of hits is always one quarter of your picks (10 of 40 are drawn), so the most likely outcome is the nearest whole number below that. And more picks do not make the middle outcomes better, they make the extremes rarer: with 10 picks, 8 hits is 1 in 43,303, 9 hits is 1 in 2.8 million, and all 10 is 1 in 847,660,528.

How the pay tables work

The game has a pay table for each number of picks and each risk level. It is built in three steps:

1. A minimum number of hits to pay. Low risk pays from 25% of your picks (rounded up, at least 1 hit), medium from 40%, high from 50%.
2. A shape. Each hit count's payout is made proportional to a power of how rare it is. The higher the risk, the steeper the climb toward the rarest hits.
3. A scale. All the payouts are multiplied by one number so that the whole table returns exactly 97%.

Here are the results for 5 picks:

Check the low-risk column: 27.77% x 1.8981 + 7.93% x 3.7807 + 0.96% x 12.0965 + 0.038% x 71.044 = 0.527 + 0.300 + 0.116 + 0.027 = 0.970. Every table in the game passes that test, whatever the picks and risk.

For 10 picks, the top end hits the game-wide cap of 50,000 times the bet:

The cap matters here. No round in Cashout Arcade pays more than 50,000 times the bet, and a plain clamp would drop the 10-pick high-risk return from 97% to about 66%, since most of that table's return sits in the top hits. So the game rebuilds the rest of the table to compensate: the payouts for 5 to 8 hits are rescaled so that the table still returns exactly 97%. In other words, a 50,000x prize for a 1-in-2.8-million hit is what the table can afford.

Why every pick count returns 97%

Every table is scaled to 97%, so there is no best number of picks for the return. What the pick count changes is how the return is distributed:

One pick is a simple 25% chance paying 3.88x (that is 0.97 divided by 0.25). With more picks you trade frequent small outcomes for rare large ones. At one pick the risk setting does nothing at all, because there is only one result that can pay.

A practical rule: if you want the game to last, choose a low risk level with 3 to 5 picks, which pays something (usually modestly) 37% to 70% of the time. If you want the occasional large result, choose a high risk level and accept that most rounds return nothing. See the volatility guide for why those two can have the same return.

Hot and cold numbers

There are no hot or cold numbers. Every number from 1 to 40 has a 25% chance in every round, the draws are independent, and the game does not remember earlier rounds. The "best numbers" to pick are the ones you like, because any set of k numbers has exactly the same hit distribution as any other. Birthdays, repeating a ticket or choosing "overdue" numbers makes no difference. The only effect the numbers can have is on your enjoyment.

If a number has not appeared for 20 rounds, that has a probability of 0.75 to the power of 20, roughly 0.3%, so it is rare for a given number, but with 40 numbers some number will go that long fairly often. See the streaks guide.

Related games

Duck Pond and Lightning began as keno-style games with the same 40-number idea. In the current app each has its own rules and payout table, so do not apply the tables above to them: Duck Pond has you pick ducks from a fixed pond, and Lightning has you pick buildings in a storm. Their game pages list their own odds. The classic keno game is on the Keno page, and every game's return is on the Odds page.