Tuesday December 23, 2008

Guest Column

Club Keno: Taxing People that suck at Math

Breaking down the mystery of the sweet science: Club Keno's true odds

By Jon Boland


As many of you already know, Club Keno is a lottery game that may be played at fine watering holes wherever lotteries are legal.  I was first introduced to Club Keno shortly after moving to Michigan at Paul Revere’s Tavern in East Lansing.  Everyone at the table had his own theory of which Keno game to play (that is, how many numbers to choose), and which numbers were “hot”, meaning that the numbers were selected more often than other numbers.


Being a “numbers guy” I wanted to learn the math behind the game so that I could get the best chance of winning a little cash the next time I played Club Keno.  But, I’ve never been good at calculating probabilities.  So, I searched on the Internet to find the equation used to calculate the odds of winning at Club Keno.  Here’s an excerpt from www.robohoo.com:


The proper buzzword for keno odds is "hypergeometric distribution". But as usual, understanding the math is far less important than understanding how to apply it properly. First, let's do the basics: if you mark N spots, the probability of hitting exactly K of them is given by the formula:



The expression C(X,Y) represents the number of possible ways to select Y items from a larger collection of X items, where order of selection is unimportant. Many calculators, spreadsheets and math libraries have a built-in facility for calculating this function. Both Lotus 1-2-3 ™ and Excel ™ name this funcion COMBIN(n,r); it is also known as the "binomial coefficient" function. (Caution: even if defined by your spreadsheet you may find the numbers involved too large to be handled by your spreadsheet program). Direct evaluation comes from the following formula:



... where "X!", pronounced "X factorial", is the product of all whole numbers from 1 to X. Thus 4! = 1 x 2 x 3 x 4 = 24. As a degenerate case, 0! = 1. So C(5,3) = 5!/(3!x2!) = 120/12 = 10. There are 10 ways to select 3 items from a bag of 5 items. Again, order of selection is unimportant. Note that N! = N x (N-1)!, for N>0. This can be useful in simplifying calculation.


I used this information to build a spreadsheet that shows the probability of matching any possible Keno combination.  I then converted the probabilities to odds of matching.  The results of these calculations are in the following tables.

Using the probabilities and the payouts for each combination, I calculated the expected return for each dollar wagered, as shown in the following tables.

Obviously, the odds are not in your favor.  In fact, you’re going to lose just about every time (across the board, the overall odds of losing are a little worse than one in one).  However, the results in the Risk Adjusted Returns Table may surprise some, and debunk a few of those theories.  Certainly the goal of any game of chance is to maximize your expected return, or as is generally the case, minimize your expected loss.  The calculated expected loss is about the same for all of the Keno games, a $0.35 loss, with the exception of the Pick 1 game, which has an expected loss of $0.50.  The game with the smallest expected loss is the Pick 2 game.  Interestingly enough, the game designers have disguised this fact by only giving us the overall odds of winning, or as I call them, the overall odds of NOT losing (NL in the table), since several of the “winning” combinations simply have you breaking even.  The Pick 2 game has the worst overall odds, but regardless of the odds, it has the smallest expected loss.  One theory postulated the first time I played was that the Pick 4 game was the best because it has the best overall odds of “winning”.  But note in the Odds Table that the odds of breaking even in the Pick 4 game are worse than the odds of getting paid in the Pick 1 game!  The actual overall odds of getting paid in the Pick 4 game are one in 21.59.  The Pick 1 game actually has the best odds of getting paid, but the payout stinks.


The moral of the story is that you should always expect to lose money playing Club Keno.  However, you’ll lose the least amount of money and stretch your entertainment dollars a tad further if you play the Pick 2 game.  Combine that with your “hot numbers” theories, which are of course statistically bunk, and who knows what you can do!