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 toคาสิโนbets. 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.

Where to Play

In an effort to judge which case has the best keno I did a comparison of the expected return of the pick 9. The following table presents my results in order or return, from highest to lowest.

Pick 9 Return

Casino Return

Silverton 79.85%

Arizona Charlie’s 75.13%

Frontier 74.83%

Jerry’s Nugget 74.78%

Nevada Palace 74.62%

Orleans 74.39%

Gold Coast 74.39%

Sam’s Town 74.28%

Las Vegas Club 72.82%

Rio 72.76%

Mirage 71.87%

Bellagio 71.87%

Eldorado (Henderson) 71.38%

Golden Nugget 71.38%

MGM Grand 71.13%

New York New York 71.13%

Primm Valley Resorts 70.86%

Hilton 70.8%

Fitzgeralds 70.8%

Western 70.8%

Sahara 70.8%

Western 70.35%

Luxor 70.23%

Circus Circus 70.23%

Main Street Station 70.12%

California 70.12%

Riviera 69.66%

Stardust 69.44%

Plaza 69.18%

San Remo 69.08%

Aladdin 68.52%

Fremont 68.52%

Four Queens 68.52%

Bally’s 68.17%

Treasure Island 67.54%

Caesars Palace 67.54%

Station Casinos 66.54%

Palms 66.24%

Monte Carlo 65.26%

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.

Pick Table 1 Table 2 Table 3 Table 4 Table 5 Table 6 Table 7 Table 8

1 75% 75% 75% 75% 75% 75%

2 72.15% 90.19% 84.18% 90.19% 84.18% 90.19% 90.19% 90.19%

3 72.15% 94.35% 86.03% 91.58% 83.25% 87.41% 91.58% 92.96%

4 72.87% 94.78% 86.14% 92.03% 86.14% 87.74% 92.03% 92.77%

5 71.93% 94.95% 85.96% 91.93% 85.31% 88.06% 91.93% 93.33%

6 70.73% 94.99% 85.88% 92.67% 85.21% 88.02% 92.67% 92.66%

7 69.73% 94.92% 86.04% 92.44% 85.31% 87.68% 92.44% 92.64%

8 70.04% 94.9% 86.17% 92.31% 84.17% 88.2% 92.31% 92.62%

9 70.8% 93.6% 85.8% 92.39% 84.87% 87.57% 92% 92.66%

10 70.33% 93.2% 85.81% 92.75% 86.72% 88.8% 92.55% 92.69%

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.