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# Gaming Guru

### Keno

13 February 2005

Judging from the sizes of Keno lounges, the casino game of Keno is not as popular with casino players as it once was. Still many people play the game and many, many more are now playing it in connection with state lotteries. In my own state (Massachusetts) one can play Keno at selected bar rooms every four minutes from 5:00 PM until the bar closes. For those few of you who might not know how it is played let me explain the rules; I'll use the six spot game to illustrate them.

The player selects six numbers from the numbers 1 through 80 and marks them on a special Keno card. The player then takes the card to a lottery representative (a Keno runner, an official in the Keno lounge, or, in the case of state lotteries, a lottery representative located in the bar). The numbers are entered into the computer and the player receives a receipt confirming his number selection.

The Keno game then randomly selects 20 of the numbers 1 through 80 and displays them on an eight by ten grid. In casinos this selection is made using a ping pong ball blower similar to that used in Bingo games and in lottery games the selection is made using a random number generator.

If enough of the player's numbers match the 20 selected by the Keno game then the player wins a cash prize. Here is the payout schedule for the Massachusetts 6 spot Keno game:

 Number of Matches Payoff 6 1600 5 50 4 7 3 1 0 through 2 0

Figure 1
Typical Keno Payout

Is this a good deal? Let's find out using some of the ideas we learned in last month's article.

It will be convenient to imagine that the Keno game is played a bit differently than it really is. I want to suppose that the 20 Keno numbers are picked prior to the player's 6 picks but the player doesn't know what they are until they are displayed on the Keno board. Clearly this is equivalent to the game as it is actually played.

Very well, once the 20 Keno numbers are selected the 80 numbers are partitioned into two sets, the twenty that I'll dub winners and the remaining 60 that I'll dub losers. If we want to know how many ways there are to pick six numbers that consist of, say four winners and two losers, this would just be the number of ways to pick four out of twenty times the number of ways to pick two out of sixty. Using our results from last month's article, this would simply be C(20,4) x C(60,2). Using the formula we derived for C(n,k) this is just

C(20,4) x C(60,2) = 4,845 x 1,770 = 8,575,650

The other outcomes are calculated in an analogous manner. The results are tabulated below

 Symbols Calculation Result C(20,6) x C(60,2) 38,760 x 1 38,760 C(20,5) x C(60,1) 15,504 x 60 930,240 C(20,4) x C(60,2) 4,845 x 1,770 8,575,650 C(20,3) x C(60,3) 1,140 x 34,220 39,010,800 C(20,2) x C(60,4) 190 x 487,635 92,650,650 C(20,1) x C(60,5) 20 x 5,461,512 109,230,240 C(20,0) x C(20,6) 1 x 50,063,860 50,063,860 Total - --- 300,500,200

Figure 2
Keno Calculations

Observe that the total number of ways to pick six numbers out of 80 is just C(80,6) and if you calculate that number it is 300,500,200, exactly the same as the total in Figure 2. This is a good way to check that our calculations are correct. Here is another interesting fact. From 5:00 PM to 1:00 AM (closing time) there are 120 four-minute intervals. If you divide the total number of combinations by 120 and then divide the result by 365, it turns out that you alone would have to play Keno for approximately 6,861 years to play every six number combination.

We are now in a position to analyze Keno. Suppose that 300,500,200 people bet a dollar each, each person having a different six-number combination from any of the other players. Clearly the Keno operator (casino or lottery) would collect \$300,500,200. How much of this would be returned to the players. From figures 1 and 2 we can construct the table shown in Figure 3. Here is the explanation of how the table is formed. 38,760 of the players will match six numbers and each then gets \$1600 back. So, the total returned in this case would be the product of 38,760 times \$1600 which is \$62,016,000. The same type of calculation holds for each of the other matches. The figures under the Product column, therefore, represent monies returned to the players. The sum, which is shown in the lower right corner, represents the total returned to the players.

 Matches Frequency Payback Product 6 38,760 1600 62,016,000 5 930,240 50 46,512,000 4 8,575,650 7 60,029,550 3 39,010,800 1 39,010,800 2 92,650,650 0 0 1 109,230,240 0 0 0 50,063,860 0 0 Total - 300,500,200 --- 207,568,350

Figure 3
Total Return Calculation

The difference between the money wagered and the amount returned is the amount that is kept by the Keno operator; that figure is \$92,931,850. This represents approximately 30.926% of the amount wagered and is the house edge on this game. It is huge! Clearly the Massachusetts Keno game is one to avoid. What about other Keno games?

Suffice it to say that some are better than others but they are all bad. In his January 2005 edition of the Las Vegas Advisor, Anthony Curtis quotes Charles Lund as reporting that the El Cortez in downtown Las Vegas offers a Keno game that returns 85.9% to the players. This is the highest return I've heard of, but it is still poor.

Although I used the six-spot game to illustrate the Keno calculation, any Keno game can be analyzed using the ideas above. The payouts are obtained from a Keno brochure and the frequencies can be calculated as I did in Figure 2. Here is a tip though. For games with a lot of spots you will run into some huge numbers. If you want to see how to keep the numbers manageable just write to me at 711cat@comcast.net and I'll give you the details.

See you next month.

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Best of Donald Catlin
Donald Catlin
Don Catlin is a retired professor of mathematics and statistics from the University of Massachusetts. His original research area was in Stochastic Estimation applied to submarine navigation problems but has spent the last several years doing gaming analysis for gaming developers and writing about gaming. He is the author of The Lottery Book, The Truth Behind the Numbers published by Bonus books.

#### Books by Donald Catlin:

Lottery Book: The Truth Behind the Numbers
Donald Catlin
Don Catlin is a retired professor of mathematics and statistics from the University of Massachusetts. His original research area was in Stochastic Estimation applied to submarine navigation problems but has spent the last several years doing gaming analysis for gaming developers and writing about gaming. He is the author of The Lottery Book, The Truth Behind the Numbers published by Bonus books.

#### Books by Donald Catlin:

Lottery Book: The Truth Behind the Numbers