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.
Basically, any sort of
Keno Strategies are based simply on mathematical computations,
such as those below.
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
.
*The following online
casinos are highly reputable, and we recommend them for trying
out our keno strategies, whether you want to play for
free/fun, or if you want to play for real money. Good
luck!
