To understand keno probabilities you must first fully understand the combinatorial function. For example, in the Maryland lotto the player picks 6 numbers out of 49. Then the lottery will draw 6 numbers out of 49, without replacement. The player wins the jackpot if all six numbers match (order does not matter). If you don’t know the probability of winning per game is 1 in 13,983,816 then you need to review the combinatorial function before going further.

In Online Keno the casino, or game machine, will draw 20 numbers out of 80, without replacement. Before this happens the player may pick 1 to 15 (or more) numbers. A "catch" is a number the player picked and was drawn by the casino. A "miss" is a number of the player picked but was not drawn by the casino. The player is paid according to the number of picks made and catches.

Although the player picks first the combinations get very huge when calculation the number of ways the casino can match the player’s picks. To start with there are combin(80,20) = 3,535,316,142,212,180,000 ways the casino can draw 20 numbers out of 80.

Many calculators don’t support numbers this big. So to keep the combinations smaller we will assume the casino draws first, but keeps the numbers secret, and player tries to match the numbers already drawn. I assure you the math is the same either way.

All the following images were taken from the Luxor keno rulebook. All pays are on a "for one" basis. In other words the player never gets his original bet back, even if he wins.

For any bet the expected return is the dot product over every possible event of the probability and what it pays. The following table shows every possible outcome, with the total in the bottom row.