From: ov@head-cfa.harvard.edu (Olaf Vancura)
Newsgroups: rec.gambling
Subject: Keno payoffs and returns...
Date: 17 Dec 92 16:03:35 GMT
Distribution: usa
Organization: Smithsonian Astrophysical Observatory, Cambridge, MA,  USA

Below are the calculated results of playing Keno.

Columns are 1) Number of spots picked.
 	    2) Number of spots caught.
	    3) Probability of catching this number.
	    4) Payoff for this number caught (from Riverside in
	       Reno, but very similar to others).
            5) Expected return for that number caught.
Finally, the average return for each dollar bet.

For example, with 3 spots picked, you have a 13.9% chance of
catching two, with a payoff of $1 (i.e. you get your money back).
Betting 3 spots, you can expect, on average, to win only 73.5 cents
for every dollar bet.

Some trends to notice:  The best bet to make is the single spot,
with a return of only 75%!!  This should indicate just how poor a bet
keno really is!  The easiest way to win $100,000 is to correctly
pick 10 out of 10, with a probability of 1.12E-7.  To put this in
context, if you played a round of Keno every five minutes, after 10
years of playing (1,000,000 games), you would still only have roughly 
a 10% chance of having hit 10 out of 10 (and in the meantime would have
lost about $282,000!!).  Or, consider the odds of getting 15 out of 15.
If you played a game of keno every second of every minute of every hour, 
etc., for 1000 years, your chance of having hit 15 out of 15 would be 
only 7%!!  Aha, but somebody has to pay for all those flashing lights 
in the casinos...

If you want the game to last (although by bucking 25% your bankroll
won't last long) you ought to bet only 1 spot, which has the highest
return.  If you want the greatest chance of "winning" anything, bet
four spots, since about 26% of the time you'll have some money coming
back to you.

Incidentally, if you have other payoff schedules, you can calculate the
return by multiplying the chance of catching a particular number (column 3
below) by your payoff (insert into column 4) to get the return on that 
number caught (multiply columns 3 and 4 to get column 5).  
Then sum up column 5 to get the total return.

Presently the FAQ has no section on Keno, but perhaps this should go in 
there somewhere to impress upon people the poor game that it is, or
from the casino's point of view, what a windfall it is...

Here are the numbers:

   1   0  7.50E-01  0.00E+00  0.00E+00
   1   1  2.50E-01  3.00E+00  7.50E-01
Total Return for each 1.0 bet:            7.50E-01

   2   0  5.60E-01  0.00E+00  0.00E+00
   2   1  3.80E-01  0.00E+00  0.00E+00
   2   2  6.01E-02  1.20E+01  7.22E-01
Total Return for each 1.0 bet:            7.22E-01

   3   0  4.17E-01  0.00E+00  0.00E+00
   3   1  4.31E-01  0.00E+00  0.00E+00
   3   2  1.39E-01  1.00E+00  1.39E-01
   3   3  1.39E-02  4.30E+01  5.97E-01
Total Return for each 1.0 bet:            7.35E-01

   4   0  3.08E-01  0.00E+00  0.00E+00
   4   1  4.33E-01  0.00E+00  0.00E+00
   4   2  2.13E-01  1.00E+00  2.13E-01
   4   3  4.32E-02  3.00E+00  1.30E-01
   4   4  3.06E-03  1.25E+02  3.83E-01
Total Return for each 1.0 bet:            7.25E-01

   5   0  2.27E-01  0.00E+00  0.00E+00
   5   1  4.06E-01  0.00E+00  0.00E+00
   5   2  2.70E-01  0.00E+00  0.00E+00
   5   3  8.39E-02  1.00E+00  8.39E-02
   5   4  1.21E-02  1.20E+01  1.45E-01
   5   5  6.45E-04  7.50E+02  4.84E-01
Total Return for each 1.0 bet:            7.13E-01

   6   0  1.67E-01  0.00E+00  0.00E+00
   6   1  3.63E-01  0.00E+00  0.00E+00
   6   2  3.08E-01  0.00E+00  0.00E+00
   6   3  1.30E-01  1.00E+00  1.30E-01
   6   4  2.85E-02  4.00E+00  1.14E-01
   6   5  3.10E-03  9.50E+01  2.94E-01
   6   6  1.29E-04  1.50E+03  1.93E-01
Total Return for each 1.0 bet:            7.32E-01

   7   0  1.22E-01  0.00E+00  0.00E+00
   7   1  3.15E-01  0.00E+00  0.00E+00
   7   2  3.27E-01  0.00E+00  0.00E+00
   7   3  1.75E-01  0.00E+00  0.00E+00
   7   4  5.22E-02  1.50E+00  7.83E-02
   7   5  8.64E-03  2.00E+01  1.73E-01
   7   6  7.32E-04  4.25E+02  3.11E-01
   7   7  2.44E-05  6.00E+03  1.46E-01
Total Return for each 1.0 bet:            7.09E-01

   8   0  8.83E-02  0.00E+00  0.00E+00
   8   1  2.66E-01  0.00E+00  0.00E+00
   8   2  3.28E-01  0.00E+00  0.00E+00
   8   3  2.15E-01  0.00E+00  0.00E+00
   8   4  8.15E-02  0.00E+00  0.00E+00
   8   5  1.83E-02  7.00E+00  1.28E-01
   8   6  2.37E-03  9.00E+01  2.13E-01
   8   7  1.60E-04  1.50E+03  2.41E-01
   8   8  4.35E-06  2.80E+04  1.22E-01
Total Return for each 1.0 bet:            7.03E-01

   9   0  6.37E-02  0.00E+00  0.00E+00
   9   1  2.21E-01  0.00E+00  0.00E+00
   9   2  3.16E-01  0.00E+00  0.00E+00
   9   3  2.46E-01  0.00E+00  0.00E+00
   9   4  1.14E-01  5.00E-01  5.71E-02
   9   5  3.26E-02  3.00E+00  9.78E-02
   9   6  5.72E-03  4.40E+01  2.52E-01
   9   7  5.92E-04  2.80E+02  1.66E-01
   9   8  3.26E-05  3.50E+03  1.14E-01
   9   9  7.24E-07  3.50E+04  2.53E-02
Total Return for each 1.0 bet:            7.12E-01

  10   0  4.58E-02  0.00E+00  0.00E+00
  10   1  1.80E-01  0.00E+00  0.00E+00
  10   2  2.95E-01  0.00E+00  0.00E+00
  10   3  2.67E-01  0.00E+00  0.00E+00
  10   4  1.47E-01  0.00E+00  0.00E+00
  10   5  5.14E-02  1.00E+00  5.14E-02
  10   6  1.15E-02  2.00E+01  2.30E-01
  10   7  1.61E-03  1.50E+02  2.42E-01
  10   8  1.35E-04  1.00E+03  1.35E-01
  10   9  6.12E-06  8.00E+03  4.90E-02
  10  10  1.12E-07  1.00E+05  1.12E-02
Total Return for each 1.0 bet:            7.18E-01

  11   0  3.27E-02  0.00E+00  0.00E+00
  11   1  1.44E-01  0.00E+00  0.00E+00
  11   2  2.68E-01  0.00E+00  0.00E+00
  11   3  2.78E-01  0.00E+00  0.00E+00
  11   4  1.79E-01  0.00E+00  0.00E+00
  11   5  7.41E-02  1.00E+00  7.41E-02
  11   6  2.02E-02  8.00E+00  1.62E-01
  11   7  3.61E-03  7.00E+01  2.53E-01
  11   8  4.11E-04  3.60E+02  1.48E-01
  11   9  2.84E-05  2.00E+03  5.67E-02
  11  10  1.06E-06  1.00E+04  1.06E-02
  11  11  1.60E-08  1.00E+05  1.60E-03
Total Return for each 1.0 bet:            7.05E-01

  12   0  2.32E-02  0.00E+00  0.00E+00
  12   1  1.14E-01  0.00E+00  0.00E+00
  12   2  2.38E-01  0.00E+00  0.00E+00
  12   3  2.80E-01  0.00E+00  0.00E+00
  12   4  2.06E-01  0.00E+00  0.00E+00
  12   5  9.94E-02  1.00E+00  9.94E-02
  12   6  3.22E-02  4.00E+00  1.29E-01
  12   7  7.03E-03  2.80E+01  1.97E-01
  12   8  1.02E-03  2.20E+02  2.24E-01
  12   9  9.54E-05  5.50E+02  5.25E-02
  12  10  5.43E-06  2.00E+03  1.09E-02
  12  11  1.67E-07  2.00E+04  3.35E-03
  12  12  2.09E-09  1.00E+05  2.09E-04
Total Return for each 1.0 bet:            7.16E-01

  13   0  1.64E-02  0.00E+00  0.00E+00
  13   1  8.88E-02  0.00E+00  0.00E+00
  13   2  2.07E-01  0.00E+00  0.00E+00
  13   3  2.73E-01  0.00E+00  0.00E+00
  13   4  2.27E-01  0.00E+00  0.00E+00
  13   5  1.26E-01  0.00E+00  0.00E+00
  13   6  4.75E-02  2.00E+00  9.50E-02
  13   7  1.23E-02  1.60E+01  1.97E-01
  13   8  2.18E-03  7.00E+01  1.53E-01
  13   9  2.60E-04  7.00E+02  1.82E-01
  13  10  2.01E-05  4.00E+03  8.02E-02
  13  11  9.43E-07  7.00E+03  6.60E-03
  13  12  2.40E-08  2.50E+04  6.00E-04
  13  13  2.46E-10  1.00E+05  2.46E-05
Total Return for each 1.0 bet:            7.14E-01

  14   0  1.15E-02  0.00E+00  0.00E+00
  14   1  6.85E-02  0.00E+00  0.00E+00
  14   2  1.76E-01  0.00E+00  0.00E+00
  14   3  2.59E-01  0.00E+00  0.00E+00
  14   4  2.42E-01  0.00E+00  0.00E+00
  14   5  1.52E-01  0.00E+00  0.00E+00
  14   6  6.58E-02  1.00E+00  6.58E-02
  14   7  1.99E-02  1.00E+01  1.99E-01
  14   8  4.18E-03  4.00E+01  1.67E-01
  14   9  6.08E-04  3.00E+02  1.82E-01
  14  10  5.97E-05  1.00E+03  5.97E-02
  14  11  3.81E-06  6.00E+03  2.29E-02
  14  12  1.48E-07  2.50E+04  3.70E-03
  14  13  3.08E-09  7.50E+04  2.31E-04
  14  14  2.57E-11  1.00E+05  2.57E-06
Total Return for each 1.0 bet:            7.01E-01

  15   0  8.02E-03  0.00E+00  0.00E+00
  15   1  5.23E-02  0.00E+00  0.00E+00
  15   2  1.48E-01  0.00E+00  0.00E+00
  15   3  2.40E-01  0.00E+00  0.00E+00
  15   4  2.50E-01  0.00E+00  0.00E+00
  15   5  1.76E-01  0.00E+00  0.00E+00
  15   6  8.63E-02  1.00E+00  8.63E-02
  15   7  2.99E-02  6.00E+00  1.79E-01
  15   8  7.33E-03  3.00E+01  2.20E-01
  15   9  1.27E-03  1.00E+02  1.27E-01
  15  10  1.52E-04  3.00E+02  4.56E-02
  15  11  1.23E-05  2.50E+03  3.09E-02
  15  12  6.50E-07  7.00E+03  4.55E-03
  15  13  2.07E-08  2.50E+04  5.17E-04
  15  14  3.50E-10  7.50E+04  2.63E-05
  15  15  2.34E-12  1.00E+05  2.34E-07
Total Return for each 1.0 bet:            6.94E-01

Olaf Vancura
ov@cfa257.harvard.edu
Newsgroups: rec.gambling