Editing Hypergeometric distribution (section)

=== Application to Keno ===
The hypergeometric distribution is indispensable for calculating [[Keno]] odds. In Keno, 20 balls are randomly drawn from a collection of 80 numbered balls in a container, rather like [[Bingo (American version)|American Bingo]]. Prior to each draw, a player selects a certain number of ''spots'' by marking a paper form supplied for this purpose. For example, a player might ''play a 6-spot'' by marking 6 numbers, each from a range of 1 through 80 inclusive. Then (after all players have taken their forms to a cashier and been given a duplicate of their marked form, and paid their wager) 20 balls are drawn. Some of the balls drawn may match some or all of the balls selected by the player. Generally speaking, the more ''hits'' (balls drawn that match player numbers selected) the greater the payoff.

For example, if a customer bets ("plays") $1 for a 6-spot (not an uncommon example) and hits 4 out of the 6, the casino would pay out $4. Payouts can vary from one casino to the next, but $4 is a typical value here. The probability of this event is:
:<math> P(X=4) = f(4;80,6,20) = {{{6 \choose 4} {{80-6} \choose {20-4}}}\over {80 \choose 20}} \approx 0.02853791</math>

Similarly, the chance for hitting 5 spots out of 6 selected is
<math> {{{6 \choose 5} {{74} \choose {15}}} \over {80 \choose 20}} \approx 0.003095639</math>
while a typical payout might be $88. The payout for hitting all 6 would be around $1500 (probability ≈ 0.000128985 or 7752-to-1). The only other nonzero payout might be $1 for hitting 3 numbers (i.e., you get your bet back), which has a probability near 0.129819548.

Taking the sum of products of payouts times corresponding probabilities we get an expected return of 0.70986492 or roughly 71% for a 6-spot, for a house advantage of 29%. Other spots-played have a similar expected return. This very poor return (for the player) is usually explained by the large overhead (floor space, equipment, personnel) required for the game.