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Keno - Casino Winning Systems and Strategies

KENO Keno Table
If you think you might get rich playing keno, you're right ... and you're wrong. To understand how you can be at both ends of the correctness spectrum, you¹ll get some help if you know the origins of the gambling game.

Keno's ancestry traces back more than 2,000 years to China, when the Chou Dynasty was beginning to crumble. An inventive individual named Cheung Leung came up with a lottery game designed to raise money for a provincial army. Leung's concept was so successful that he was able to mobilize the army successfully and to unify China under the new Han Dynasty (talk about a big-money house game!) In the years that followed, all sorts of similar lotteries appeared in China -- some legal and some not.

Played throughout the centuries in every province, a variation of Leung's game eventually found its way to the Chinese communities in America in the 1800s. There it became known, appropriately, as the Chinese lottery.

In 1931, the game was introduced into the legal casinos of Nevada under the name racehorse keno. Gamers thinly disguised it as a bingo-type game since bingo was legal and lotteries were not. Over the years, undergoing several changes, it became known as keno.

Now, just knowing that the game uses a lottery as its basis should be a big clue as to why you probably won't get rich playing it. After all, Cheung Leung managed scrape off an edge big enough to support the largest country in the world. However, knowing that the game uses a lottery as its basis should also be a big clue as to why you could get rich playing it. Haven't we all read about those instant multimillionaires, thanks to Powerball? (Fortunately, today's keno game uses fewer numbers than the original version, so the house edge isn't as large as it was in Leung's day.)

THE GAME
Keno as we know it today consists of a paper ticket imprinted with the numbers 1 through 80.

You, the player, select favourite or lucky numbers, mark these numbers with an X using a crayon provided by the casino, determine how much money you want to invest in the game, then submit the ticket to a keno writer or runner. How much you can win is determined by the number of "hits" (matching numbers) you get -- and how much money you invested. Obviously, the more numbers that come in and the more money you've wagered have a lot to do with the payoff ... and the more numbers you pick, the harder it is to win.

FOR EXAMPLE
If you  hit six out of six in this casino, we'll win $1,600. If we hit five of the six numbers, we'll get back $80. Four out of six will yield us $4 while three out of six will get us our $1 back.

In days gone by, you could play just one game at a time. Today, you can play two, three, even 100 games, just by indicating that preference on your ticket and putting up the appropriate amount of cash. At some casinos, you can play 1,000 games and have up to a year to present your ticket for validation and, if necessary, collection of winnings.

You can play a one-spot, a two-spot, all the way up (in some casinos) to a 15-spot ticket, and your results, should you hit, vary according to the casino.

KENO WINNING SYSTEMS
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 0.72151898734170

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 0.12157425195400 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
Total 1.00000000000000 0.71960087466417

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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Keno - Casino Winning Systems and Strategies

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