Editing Keno (section)

==Probabilities==
Keno payouts are based on how many numbers the player chooses and how many of those numbers are "hit", multiplied by the proportion of the player's original wager to the "base rate" of the paytable. Typically, the more numbers a player chooses and the more numbers hit, the greater the payout, although some paytables pay for hitting a lesser number of spots. For example, it is not uncommon to see casinos paying $500 or even $1,000 for a “catch” of 0 out of 20 on a 20 spot ticket with a $5.00 wager. Payouts vary widely by casino. Most casinos allow paytable wagers of 1 through 20 numbers, but some limit the choice to only 1 through 10, 12 and 15 numbers, or "spots" as keno aficionados call the numbers selected.<ref>{{cite web|url=http://www.gamblinginfo.com/18_Keno.htm|title=Tutorial - How to play Keno|publisher=Gambling Info|access-date=27 June 2011|archive-date=6 May 2021|archive-url=https://web.archive.org/web/20210506074928/https://www.gamblinginfo.com/18_Keno.htm|url-status=dead}}</ref>

The [[probability]] of a player hitting all 20 numbers on a 20 spot ticket is approximately 1 in 3.5 [[quintillion]] (1 in 3,535,316,142,212,174,320).<ref>[http://www.mathproblems.info/gam470/games/keno/prob-keno.html Probabilities in keno]</ref>

Even though it is highly improbable to hit all 20 numbers on a 20 spot ticket, the same player would typically also get paid for hitting “catches” 0, 1, 2, 3, and 7 through 19 out of 20, often with the 17 through 19 catches paying the same as the solid 20 hit. Some of the other paying "catches" on a 20 spot ticket or any other ticket with high "solid catch" odds are in reality very possible to hit:

{| class="wikitable"
! Hits !! Probability
|-
| 0    || 1 in 843.380 (0.11857057%)
|-
| 1    || 1 in 86.446 (1.15678605%)
|-
| 2    || 1 in 20.115 (4.97142576%)
|-
| 3    || 1 in 8.009 (12.48637168%)
|-
| 4    || 1 in 4.877 (20.50318987%)
|-
| 5    || 1 in 4.287 (23.32807380%)
|-
| 6    || 1 in 5.258 (19.01745147%)
|-
| 7    || 1 in 8.826 (11.32954556%)
|-
| 8    || 1 in 20.055 (4.98618021%)
|-
| 9    || 1 in 61.420 (1.62814048%)
|-
| 10   || 1 in 253.801 (0.39401000%)
|-
| 11   || 1 in 1,423.822 (0.07023351%)
|-
| 12   || 1 in 10,968.701 (0.00911685%)
|-
| 13   || 1 in 118,084.920 (0.00084685%)
|-
| 14   || 1 in 1,821,881.628 (0.00005489%)
|-
| 15   || 1 in 41,751,453.986 (0.00000240%)
|-
| 16   || 1 in 1,496,372,110.872 (0.00000007%)
|-
| 17   || 1 in 90,624,035,964.712
|-
| 18   || 1 in 10,512,388,171,906.553
|-
| 19   || 1 in 2,946,096,785,176,811.500
|-
| 20   || 1 in 3,535,316,142,212,173,800.000
|}

Probabilities change significantly based on the number of spots and numbers that are picked on each ticket.

=== Probability calculation ===

Keno probabilities come from a [[hypergeometric distribution]].<ref>{{Cite book
| edition   = Third
| publisher = Duxbury Press
| last      = Rice
| first     = John A.
| title     = Mathematical Statistics and Data Analysis
| year      = 2007
| page      = 42
}}</ref><!--  --------- -->
<ref>
{{Cite web
| url = https://statisticsbyjim.com/probability/hypergeometric-distribution/
| title = Hypergeometric Distribution: Uses, Calculator & Formula
| first1=Jim | last1=Frost 
| website= statisticsbyjim.com
| access-date=2023-03-22
}}</ref>
For Keno, one calculates the probability of hitting exactly <math>r</math> spots on an <math>n</math>-spot ticket by the formula:
:P(hitting <math>r</math> spots) <math> = {{n \choose r} \times {{80-n} \choose {20-r}} \over {80 \choose 20}}  </math> for an <math>n</math>-spot ticket. 

To calculate the probability of hitting 4 spots on a 6-spot ticket, the formula is:
:<math> P(X=4) = {{{6 \choose 4} {{80-6} \choose {20-4}}}\over {80 \choose 20}} \approx 0.02853791</math>
where <math>{n \choose r}</math> is calculated as <math>n! \over r!(n-r)!</math>, where X! is notation for X [[factorial]]. Spreadsheets have the function {{code|COMBIN(n,r)}} to calculate 
<math>{n \choose r}</math>.

To calculate "odds-to-1", divide the probability into 1.0 and subtract 1 from the result.