Keno

Last update: Apr. 1, 2008

Play begins with a keno ticket and a crayon. The keno ticket will have numbers 1 through 80 to choose from. To play the game put an X though as many number as you wish, up to 10 or 15. Some casinos also have an option to pick 20, and even one entire side of a card. For convenience the player may get a quick-pick, in which the computer will randomly pick numbers for the player. The serious player may circle groups of numbers and play each possible combination of circles on one ticket. While this may make the game more fun it does not change the massive odds against the player.

After filling out a ticket the player will take it to the counter with his money and the keno host will collect the wager and enter the picks in the computer give the player a receipt. The ticket is not proof of making a bet, only the receipt is. About a minute before the numbers are ready to be drawn the windows will be closed to new to bets. Then 80 ping pong sized balls will be whipped about an air chamber and one at a time 20 of them will be selected. If enough of the chosen 20 match the player's picks then the ticket wins according to the number of matches and the amount bet. If the tickets loses, as they usually do, then the next round is just about five minutes away.

9-Spot Survey

In 2001 I did a comparitive study of pick-9 keno games in Las Vegas. The following table presents my results in order or return, from highest to lowest. Since I did this survey, keno has been on the decline, and many of these casinos took out their live keno games.

Here is a good link on the expected returns of lots of keno games: www.johnph77.com/math/kenopct.html (very large file).

Keno Bargains

El Cortez

My keno scout "P.K." informs me that the El Cortez has a relatively competitive pay table on their $1.15 special rate. The following table shows what each hand pays as well as the return and hit frequency. Although all the picks are strong, compabibly speaking, the strongest is the pick 5 at an 87.61% return.

Orleans

My keno scout "P.K." informs me of a 90.28% return on the pick-6 $25 "Super Keno High Pay Special." The game pays $25 for 3, $180 for 4, $2500 for 5, and $50000 for 6.

Video Keno

Video keno offers a much higher return than live keno, and also a much faster pace. The following table shows the return of various video keno games, along with live keno at the Las Vegas Hilton for comparision purposes.

• Table 1: Las Vegas Hilton
• Table 2: Regent - $2 machine
• Table 3: Regent - 5 cent machine
• Table 4: Horseshoe - 25 cent machine
• Table 5: Suncoast - 5 cent machine
• Table 6: Suncoast - 5 cent machine
• Table 7: Suncoast - 5c, 10c, 25c machines
• Table 8: Suncoast - 25c, 50c, $2 machines

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 combin(52,5) = 52!/(47!*5!) = 2,598,960.

The overall general formula for the probability of x matches and y marks is combin(y,x)*combin(80-y,20-x)/combin(80,20).

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 .

To determine the expected return of an overall number of picks take the dot product of the return and the probability for each number of winning catches. For example the pick 5 at the Atlantic City Tropica pays 1 for 3 catches, 10 for 4, and 800 for 5. Thus the return is 1*combin(5,3)*combin(75,17)/combin(80,20) + 10*combin(5,4)*combin(75,16)/combin(80,20) + 800*combin(5,5)*combin(75,15)/combin(80,20) = 0.72079818915262.

Appendices

Keno appendix 1 features an analysis of the top/bottom/left/right as as well as edge tickets as played at the Las Vegas Hilton.
Keno appendix 2 features an analysis of Caveman Keno as played on IGT's Game King.
Keno appendix 3 probabilities and returns for basic keno at the Atlantic City Tropicana.
Keno appendix 4 progressive keno at the Orleans. Learn when the meter is high enough to have a positive expectation.
Keno appendix 5 Nevada numbers, a progressive lottery game at some of the Nevada Park Place casinos.
Keno appendix 6 Australian Power Keno.
Keno appendix 7 Super Keno.
Keno appendix 8 Top Bottom Keno.
Keno appendix 9High/Low/Middle Keno: Keno game seen at the Lisboa in Macau.
Keno appendix 10Extra Draw Keno.
Keno appendix 11Jumbo Bingo Progressive game at the Station Casinos.
Keno props Miscellaneous keno bets.
Probabilities in Keno. A lesson in how to calculate keno probabilities.

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