In general the easier a game is to understand the greater the house advantage, and keno is a perfect example of this. Played in a lounge or at your restaurant table, keno involves the player choosing from 1 to 15 (sometimes 20) numbers from 1 to 80. Every five minutes or so the casino will choose 20 numbers ranging from 1 to 80. If enough of your chosen numbers match those drawn by the casino then you will win, depending on exactly how many match and the payoff table at your particular casino.

While the payoff tables will vary from one casino to another the expected return seems to always range from 70 to 80 cents per dollar bet, making keno among the worst bets in the casino. Many states outside Nevada offer keno as an alternative to lottery tickets. While I can't speak for every state Maryland keno has an expected return of about 50 cents per dollar bet. I believe other state run keno to be just as bad.

Below are 15 tables, according to the number of numbers chosen, and the probability of matching any given number, the payoff table at the Atlantic City Tropicana, the contribution toward the expected return, and the total expected return for all possible matches. Following the tables is an explanation of how the probabilities were calculated.
 

Tables

 

Pick 6 Catches Pays Probability Return 0 0 0.16660175267770 0.00000000000000 1 0 0.36349473311499 0.00000000000000 2 0 0.30832142541003 0.00000000000000 3 1 0.12981954754107 0.12981954754107 4 4 0.02853791777842 0.11415167111370 5 95 0.00309563853868 0.29408566117427 6 1500 0.00012898493911 0.19347740866728 Total   1.00000000000000 0.73153428849631

 

Pick 7 Catches Pays Probability Return 0 0 0.12157425195400 0.00000000000000 1 0 0.31519250506592 0.00000000000000 2 0 0.32665405070468 0.00000000000000 3 0 0.17499324144894 0.00000000000000 4 1 0.05219096674793 0.05219096674793 5 25 0.00863850484104 0.21596262102591 6 350 0.00073207668144 0.25622683850532 7 8000 0.00002440255605 0.19522044838501 Total   1.00000000000000 0.71960087466417

 

Pick 8 Catches Pays Probability Return 0 0 0.08826623772003 0.00000000000000 1 0 0.26646411387178 0.00000000000000 2 0 0.32814562171247 0.00000000000000 3 0 0.21478622512089 0.00000000000000 4 0 0.08150370149677 0.00000000000000 5 9 0.01830258559927 0.16472327039346 6 90 0.00236671365508 0.21300422895706 7 1500 0.00016045516306 0.24068274458425 8 25000 0.00000434566067 0.10864151665261 Total   1.00000000000000 0.72705176058740

 

Pick 9 Catches Pays Probability Return 0 0 0.06374783835335 0.00000000000000 1 0 0.22066559430007 0.00000000000000 2 0 0.31642613522274 0.00000000000000 3 0 0.24610921628435 0.00000000000000 4 0 0.11410518209547 0.00000000000000 5 4 0.03260148059871 0.13040592239483 6 50 0.00571955799977 0.28597789998865 7 280 0.00059167841377 0.16566995585549 8 4000 0.00003259245500 0.13036981998314 9 50000 0.00000072427678 0.03621383888420 Total   1.00000000000000 0.74863743710631

 

Pick 10 Catches Pays Probability Return 0 0 0.04579070078903 0.00000000000000 1 0 0.17957137564325 0.00000000000000 2 0 0.29525678110572 0.00000000000000 3 0 0.26740236779386 0.00000000000000 4 0 0.14731889707162 0.00000000000000 5 1 0.05142768770500 0.05142768770500 6 22 0.01147939457701 0.25254668069420 7 150 0.00161114309853 0.24167146477914 8 1000 0.00013541935526 0.13541935526417 9 5000 0.00000612064883 0.03060324412750 10 100000 0.00000011221190 0.01122118951342 Total   1.00000000000000 0.72288962208343

 

Pick 11 Catches Pays Probability Return 0 0 0.03270764342073 0.00000000000000 1 0 0.14391363105123 0.00000000000000 2 0 0.26807441078170 0.00000000000000 3 0 0.27838496504254 0.00000000000000 4 0 0.17858658134804 0.00000000000000 5 0 0.07408035967030 0.00000000000000 6 8 0.02020373445554 0.16162987564429 7 80 0.00360780972420 0.28862477793623 8 400 0.00041141689837 0.16456675934961 9 2500 0.00002837357920 0.07093394799552 10 25000 0.00000105799787 0.02644994671019 11 100000 0.00000001603027 0.00160302707335 Total   1.00000000000000 0.71380833470919

 

Pick 12 Catches Pays Probability Return 0 0 0.02322716706690 0.00000000000000 1 0 0.11376571624603 0.00000000000000 2 0 0.23777034695421 0.00000000000000 3 0 0.27972981994613 0.00000000000000 4 0 0.20576280024883 0.00000000000000 5 0 0.09938731483717 0.00000000000000 6 5 0.03220885203057 0.16104426015283 7 32 0.00702738589758 0.22487634872249 8 200 0.00101959840032 0.20391968006364 9 1000 0.00009540101991 0.09540101991282 10 5000 0.00000542798906 0.02713994532003 11 25000 0.00000016727239 0.00418180975655 12 100000 0.00000000209090 0.00020909048783 Total   1.00000000000000 0.71677215441618

 

Pick 13 Catches Pays Probability Return 0 1 0.01639564734134 0.01639564734134 1 0 0.08880975643226 0.00000000000000 2 0 0.20661861700566 0.00000000000000 3 0 0.27273657444747 0.00000000000000 4 0 0.22728047870623 0.00000000000000 5 0 0.12587841897576 0.00000000000000 6 1 0.04750129017953 0.04750129017953 7 20 0.01231514930580 0.24630298611609 8 80 0.00218314010421 0.17465120833686 9 600 0.00025989763145 0.15593857887220 10 3500 0.00002006227331 0.07021795656818 11 10000 0.00000094336708 0.00943367083316 12 50000 0.00000002398391 0.00119919544489 13 100000 0.00000000024599 0.00002459888092 Total   1.00000000000000 0.72166513257318

 

Pick 14 Catches Pays Probability Return 0 1 0.01150142425437 0.01150142425437 1 0 0.06851912321754 0.00000000000000 2 0 0.17629399411180 0.00000000000000 3 0 0.25904423624590 0.00000000000000 4 0 0.24220636088992 0.00000000000000 5 0 0.15197261859760 0.00000000000000 6 1 0.06575738304704 0.06575738304704 7 9 0.01985128544816 0.17866156903346 8 42 0.00418163651802 0.17562873375666 9 310 0.00060823803898 0.18855379208507 10 1100 0.00005973766454 0.06571143099739 11 8000 0.00000381101528 0.03048812225484 12 25000 0.00000014784111 0.00369602775180 13 50000 0.00000000308404 0.00015420194010 14 100000 0.00000000002570 0.00000257003234 Total   1.00000000000000 0.72015525515306

 

Pick 15 Catches Pays Probability Return 0 1 0.00801614417729 0.00801614417729 1 0 0.05227920115624 0.00000000000000 2 0 0.14793901603787 0.00000000000000 3 0 0.24040090106154 0.00000000000000 4 0 0.25021318273752 0.00000000000000 5 0 0.17615008064721 0.00000000000000 6 0 0.08634807874863 0.00000000000000 7 10 0.02988971956684 0.29889719566835 8 25 0.00733144064847 0.18328601621172 9 100 0.00126716258122 0.12671625812169 10 300 0.00015205950975 0.04561785292381 11 2800 0.00001234249267 0.03455897948773 12 25000 0.00000064960488 0.01624012193972 13 50000 0.00000002067708 0.00103385391659 14 100000 0.00000000035046 0.00003504589548 15 100000 0.00000000000234 0.00000023363930 Total   1.00000000000000 0.71440170198168

Computation of Probabilities

The probability of matching x numbers, given that y were chosen, is the number of ways to select x out of y, multiplied by the number of ways to select 20-x out of 80-y, divided by the number of ways to select 20 out of 80.

The "number of ways to select x out of y" means the number of ways, without regard to order, you can select x items out of y to choose from. I shall represent this function as combin(y,x) which you can use in Excel.

For the general case combin(y,x) is y!/(x!*(y-x)!). For those of you unfamiliar with the factorial function n! is defined as 1*2*3*...*n. For example 5!=120. The number of possible five card poker hands would thus be 52!/(47!*5!) = 2,598,960.

As an example let's find the probability of getting 4 matches given that 7 were chosen. This would be the product of combin(7,4) and combin(73,16) divided by combin(80,20). combin(7,4) = 7!/(4!*3!)= 35. combin(73,16) = 73!/(16!*57!)=5271759063474610. combin(80,20) = 3535316142212170000. The probability is thus (35*5271759063474610)/3535316142212170000 =~ 0.052190967 .

Top of page