Keno odds, priced: the chance of every hit count from one pick to ten

The board, the draw, and the one formula

Keno is a lottery with the ticket price and the prize table printed on the same screen. The board has forty numbers. You mark between one and ten of them. The game draws ten balls, none of them twice, and pays according to how many of your marks were drawn. There is no decision after the draw begins and no card to hold, which means the whole game reduces to one question with an exact answer: given k marks, how likely is each number of hits?

Ten balls from forty without replacement is the hypergeometric distribution, the same arithmetic as a lottery draw. With k numbers marked, the chance of exactly h hits is C(k, h) × C(40 − k, 10 − h) ÷ C(40, 10). The denominator is the number of distinct ten-ball draws, 847,660,528, and every one of them is equally likely. Nothing in that formula mentions the risk level, and nothing in it mentions which numbers you chose. Every table below is that formula evaluated for every ticket the board allows, then cross-checked by replaying the engine’s own draw loop two million times.

The chart: every hit count for every pick count

Exact hypergeometric probabilities; each cell is an integer count of draws divided by C(40, 10) = 847,660,528. Cells below one in a million are shown as odds. Rows sum to 100%.

Read down any column and the shape is the same story told ten times. Marking more numbers shifts the weight to the right — more hits on average — but the top of every row thins out faster than the row lengthens. Two hits is the most likely result for anyone marking seven or more; one hit for three to six; and with one or two picks the most likely outcome is nothing at all.

The number worth memorising is the average. Ten of forty numbers are drawn, so each mark has exactly a one-in-four chance of being hit, and a k-pick ticket averages k ÷ 4 hits whatever else is true about it. Ten picks average 2.5 hits; four picks average one. The payout tables are all built around that average, and each one decides what to do with the ticket that lands on it.

What each pick count promises

Average hits is exactly picks ÷ 4. “At least one” is the chance the ticket touches the draw at all, which is not the same as the chance it pays — that depends on the level, below.

A ten-pick ticket almost always hits something: only one in twenty-eight comes back with nothing. That feels like a generous game, and the payout tables are where the feeling is corrected. A ten-pick ticket pays nothing below two hits on Low, below three on Classic and Medium, and below four on High, and those unpaid hit counts are the ones the chart says are common. Hitting something and being paid for it are different events, and the gap between them is how a keno table is tuned.

The last column is the one people search for. A clean sweep — every marked number drawn — is a one-in-four event with a single pick and becomes astronomically rare by ten. Each extra pick multiplies the odds against by a growing factor: the fourth pick roughly quintuples them, the tenth multiplies them by thirty-one. Five for five happens once in 2,611 tickets; eight for eight once in 1.7 million; ten for ten once in 847,660,528. The draw does not know it is being asked for a sweep, so no choice of numbers, level or timing changes any of those figures.

Hitting them all, and what the engine pays for it

Fair multiplier is 1 ÷ probability — what the all-hit cell would pay if it carried the whole return by itself. The gap to the paid multiplier is not the house edge; it is the return that the table pays out on lower hit counts instead.

High with one, two or three picks is the cleanest reading of this table, because on those rows the all-hit cell is the only cell that pays. There the multiplier is the whole return: ×3.96 against a fair ×4 is 99.00%, ×17.10 against ×17.33 is 98.65%, ×81.50 against ×82.33 is 98.99%. That is the house edge, expressed as a discount on the one prize.

Every other row pays the sweep at a fraction of its fair price, and the fraction shrinks fast. Five for five is worth ×2,611 at fair odds and pays ×450 on High, because the three- and four-hit cells on that row are paid the rest. By ten picks the ×1,000 cell is paid at about one 850,000th of its fair value, and its contribution to the ticket’s return is 1,000 ÷ 847,660,528 — 0.00012% of the stake. On a one-dollar ticket the largest number on the keno screen is worth about a ten-thousandth of a cent. The other 99% of the return is in the cells the chart calls ordinary.

When a ticket beats its stake

Counts only cells paying more than ×1. Cells that return part or all of the stake — Classic’s two ×1.00 cells (three picks with one hit, six picks with two), its ×0.80, ×0.25 and ×0.47 cells, Low’s ×0.70 and Medium’s ×0.40 with one pick and no hits — are excluded. Every figure is a hit-count probability from the chart above.

Each cell of KENO_PAYTABLE multiplied by its hit-count probability and summed. Forty tables, all within 0.42 points: the lowest is 98.65% (Medium and High at two picks), the highest 99.07% (Low at nine).

The two tables together are the whole product. The first one moves by a factor of sixty-five — Low with ten picks beats its stake on four tickets in five, High with three picks on one in eighty-three — and the second barely moves at all. Nothing you can select on the keno screen changes what a ticket is worth by more than half a point. What the selections change is how that value arrives: as a stream of small returns or as a rare large one. What a keno risk level changes takes that trade-off apart in detail; the chart above is the fixed input every one of those tables is built on.