wildbill@ucbvax.UUCP (William J. Laubenheimer) (03/13/84)
To the guy who wanted to know about keno odds a while back: Here is
a description of how to calculate them. I realize it's been a while
since you sent your request out, but I have been behind in my usenet
reading, plus a previous version of this message didn't leave UCB.
Calculating Keno Odds
As is the case with most games of chance, determining the odds
at Keno is nothing more than combinatorics plus probability theory.
In these terms, we must determine the expected value of a given
Keno ticket. Probabilistically, this is nothing more than the sum
of the products (gain of outcome X probability of outcome), where
the probability of the outcome is nothing more than the number of
cases producing the outcome divided by the total number of cases.
Here we are making the reasonable assumption that Keno is strictly
a game of chance and that all outcomes are equally likely.
Throughout the remainder of the analysis, [n r] will represent
the number of combinations of n things taken r at a time, n!/(r!(n-r)!),
and W(n, r) will represent the payoff for an n-spot ``catch r'' ticket,
in which r of the n spots chosen by the player are drawn.
The total number of draws in a Keno game is [80 20]. The number of
these which catch exactly r spots on an n-spot ticket is that which
produces r hits and (20 - r) misses. Thus, the probability of catching r is:
([n r][(80 - n) (20 - r)])/([80 20]). A little expansion into factorials
and rearranging produces a form more amenable to calculation:
([20 r][60 (n - r)])/[80 n]. This form can also be derived by considering
the drawing of balls first, and enumerating the number of tickets which
catch exactly r spots. Thus, the expected value of an n-spot Keno ticket is:
$(n) = sum from r = 0 to n of W(n, r)([20 r][60 (n - r)])/[80 n].
Now let's try some calculations for a $1.00 Keno ticket, based on
some typical Nevada payoffs:
W(n,r) r 0 1 2 3 $(n)
n
1 - 3 0.75
2 - - 12 0.7215
3 - - 1 36 0.6382
...etc. Try it for whatever case you have in mind. The same analysis
works for the variations seen at some casinos, such as payoffs for catch 0,
extra balls (20 becomes the number of balls, 60 becomes 80 - that number).
A final note on Keno odds: as you can see, they are quite bad - a
minimum of 25% to the house. This is almost twice as bad as the WORST
bet at a crap table, about 3 times as bad as even a naive blackjack player,
and 4 1/2 times as bad as roulette. Still wonder why Keno is so heavily
promoted? The only time you can get the odds in your favor is with the
coupon books many casinos give out, which often include a 50% discount
on a Keno ticket. Using one of these gives you a $.75 return on a $.50
investment -- a nice, hefty, 3 to 2 edge on the 1-spot. The 2-spot is
almost as good and has the advantage of paying a useful sum if you catch.
Back to regular Keno for a moment -- what you see is not necessarily
what you get. Down at the bottom, buried in the fine print, you will find
a line which reads something like ``aggregate limit to all players''.
What this means is that the most the casino will distribute on any game
is the limit printed on the ticket. If more than this amount is won, all
payoffs above a certain size are prorated to make sure that this limit
is not exceeded. So, if you hit a $50,000 ticket and somebody else hits
an $8,000 ticket in the same game, you probably will only get about $43,000.
If somebody else has a $50,000 ticket, you will get somewhat less than
$25,000. In short, don't play Keno with the rent money.
____ Bill Laubenheimer
___ / \ ___ UC-Berkeley Computer Science
/ \ | o o | / \ ucbvax!wildbill
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...Killjoy was here!