Keno Odds and Payouts

Marking some numbers divides the board into two groups, the marked and the unmarked, and the draw takes 20 numbers without regard to either. The chance that exactly k of the draw fall among the marked numbers is the number of ways of choosing k marked and 20 minus k unmarked, divided by the number of ways of choosing any 20 from 80. That ratio is the whole of keno's arithmetic, and every figure on this page is one instance of it.

The key number: Since the draw covers one number in four, the average ticket of four numbers lands 1 catch and the average ticket of ten lands 2.5, whatever the board looks like.

The worth of a pay-table line follows at once. If a line pays some multiple of the stake for a catch whose chance is p, the line returns p times that multiple over the long run; summed over every line, the table returns some share of the stake, and the remainder is the house edge. The right-hand column of each table is that p, which is the value of a lone 1x on the line.

The ten-pick ticket is the one most pay tables are built around. The most likely result is 2 catches, no catch at all comes once in 21.8 draws, and every number marked comes up once in 8,911,711, which is why the top line of a ten-pick table can pay thousands of times the stake and still be worth a fraction of one point.

Five numbers is where the top line is still reachable: all five come up once in 1,551 draws. A line paying the stake back on two catches would be worth 27.05% on its own, which is why so few tables pay that low a catch at all.

A small ticket settles quickly: 41.65% of draws miss every number, and a full catch of three arrives about once in 72.1 draws. Its fair payout for three catches, 72.1x, is the yardstick for what a three-pick table prints on that line.

The fair payout is the total return per unit staked at which a line would hand back exactly what it takes: one divided by the chance. A printed line below it keeps the difference. Past ten numbers the top line passes out of reach entirely, and the tables of those games pay their middle catches instead.

Take the pay table a game prints for a given pick count and the matching table above. For each line, multiply the payout by the worth of one unit on that line; add the results. The sum is the share of every stake the ticket returns over time, and one hundred per cent less that sum is the house edge. Lines the table omits contribute nothing, which is where most of the edge usually sits: on a ten-pick ticket the four results from no catch to three catches together account for most draws, and a table that pays none of them has already kept a large share before the paying lines begin.

The rule: A pay table is priced line by line: payout times the worth of that line, summed, against one hundred per cent. The chances never move; only the printed figures do, and two games with the same board can sit many points apart.

The same method tests a stated return. A game that quotes one figure usually means one pick count, and the other counts can come out above or below it; running the sum for each shows which ticket size that game treats best. It also measures a bonus ball or a multiplier draw, each a rule of its own with a chance of its own; a feature that enlarges a line worth a fraction of a point enlarges the whole by very little.

The ratio of drawn to board is unchanged here, so the average catch count per ticket is too, but each individual result moves: five of five on this board arrives once in 2,611 draws, against once in 1,551 on the full board. A game's rules screen states its board and its draw, and the figures only transfer between games that share both.