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 builtin facility for calculating this function. Both Lotus
123 ™ 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 (N1)!, 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!
