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 .

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