Keno

How To Play Keno 

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 1 Catches  Pays  Probability  Return  0  0  0.75000000000000  0.00000000000000  1  3  0.25000000000000  0.75000000000000  Total     1.00000000000000  0.75000000000000

 

Pick 2 Catches  Pays  Probability  Return  0  0  0.56012658227848  0.00000000000000  1  0  0.37974683544304  0.00000000000000  2  12  0.06012658227848  0.72151898734177  Total     1.00000000000000  0.72151898734177

 

Pick 3 Catches  Pays  Probability  Return  0  0  0.41650438169426  0.00000000000000  1  0  0.43086660175268  0.00000000000000  2  1  0.13875365141188  0.13875365141188  3  43  0.01387536514119  0.59664070107108  Total     1.00000000000000  0.73539435248296

 

Pick 4 Catches  Pays  Probability  Return  0  0  0.30832142541003  0.00000000000000  1  0  0.43273182513689  0.00000000000000  2  1  0.21263546580002  0.21263546580002  3  3  0.04324789134916  0.12974367404747  4  130  0.00306339230390  0.39824099950682  Total     1.00000000000000  0.74062013935432

 

Pick 5 Catches  Pays  Probability  Return  0  0  0.22718420819687  0.00000000000000  1  0  0.40568608606583  0.00000000000000  2  0  0.27045739071056  0.00000000000000  3  1  0.08393505228948  0.08393505228948  4  10  0.01209233804171  0.12092338041705  5  800  0.00064492469556  0.51593975644609  Total     1.00000000000000  0.72079818915262

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 .