craigGA
craigGA
Joined: Nov 29, 2009
  • Threads: 5
  • Posts: 17
October 27th, 2013 at 8:35:49 AM permalink
The Georgia Lottery's Keno game is standard Keno (20 numbers picked from 1-80). However, there are two wrinkles: (1) for double the bet, you can buy a "Multiplier". (2) Sometimes they run a "30% more" promotion where for a limited time (usually Monday 6-8PM) all winnings are increased 30%.

I THINK I have calculated the returns for all the applicable betting combinations (excluding the max payout limits)... Any corrections or feedback would be greatly appreciated.

Does this look right to you?

Pay Table and Rules:



Base Game Return (7 spot):

This portion was as simple as using The Wizards Keno Calculator:
Match Collect ($) Frequency Probability (%) Return
0 0 386206920 12.1574251954 0
1 0 1001277200 31.519250506592 0
2 0 1037687280 32.665405070468 0
3 1 555903900 17.499324144894 0.1749932414
4 3 165795900 5.219096674793 0.1565729002
5 15 27442080 0.863850484104 0.1295775726
6 125 2325600 0.073207668144 0.0915095852
7 4000 77520 0.002440255605 0.0976102242
Total 3176716400100 0.650263523681
House Edge: 34.974%

Multiplier Wheel weights:

The wheel is made up of 5 "1x" spots, 6 "2x" spots, 4 "3x" spots, 2 "5x" spots and 1 "10x" spot. I calculated the natural wheel probabilities and then downloaded 10098 trials to get the actual wheel weights (I suspected that the wheel was not naturally weighted.) Here are the results:
Multiplier Trial Count Trial Probability # of Spots Natural Probability
Mult x1 2452 0.242820360467419 5 0.277777777777778
Mult x2 5167 0.511685482273718 6 0.333333333333333
Mult x3 1740 0.172311348781937 4 0.222222222222222
Mult x5 623 0.061695385224797 2 0.111111111111111
Mult x10 116 0.0114874232521291 1 0.0555555555555556
Grand Total 10098 1 18 1

According to the 10098 trials downloaded, the wheel is not weighted naturally.

Per the official Keno rules, the overall odds of a Multiplier being 2x or greater are 1 in 1.33. The 10098 sample produced odds of 1 in 1.32069 (pretty close). I wonder if the wheel weights change slightly during the 30% promo, or if 10k samples is not enough to get closer to the 1 in 1.33 number stated by the Georgia Lottery.

Base Game Return with Multiplier and 30% Promotion (7 spot):

To calculate the impact that the Multiplier has on the return, I created a table of all possible outcomes of matched numbers and multiplier results. I then calculated returns based on the combined probability of multiplier and numbers matched. Finally, since the multiplier costs and additional $1 to play, I divide the overal return by 2 in order to get the return per dollar bet and overall house edge.

For the 30% promo, I simply added 30% to the return of each possible combination.

Numbers Matched Collect ($) Multiplier Collect Incl. Multiplier ($) Base Probability Multiplier Probability Total Probability Total Return Total Return: +30% Promo
0 0 1 0 0.121574251954 0.242820360467419 0.0295207036830271 $0.00 $0.00
0 0 2 0 0.121574251954 0.511685482273718 0.0622077797431489 $0.00 $0.00
0 0 3 0 0.121574251954 0.172311348781937 0.0209486233313488 $0.00 $0.00
0 0 5 0 0.121574251954 0.061695385224797 0.00750057030771856 $0.00 $0.00
0 0 10 0 0.121574251954 0.0114874232521291 0.00139657488875659 $0.00 $0.00
1 0 1 0 0.31519250506592 0.242820360467419 0.0765351576967356 $0.00 $0.00
1 0 2 0 0.31519250506592 0.511685482273718 0.161279428963716 $0.00 $0.00
1 0 3 0 0.31519250506592 0.172311348781937 0.0543112456738662 $0.00 $0.00
1 0 5 0 0.31519250506592 0.061695385224797 0.0194459230200107 $0.00 $0.00
1 0 10 0 0.31519250506592 0.0114874232521291 0.00362074971159108 $0.00 $0.00
2 0 1 0 0.32665405070468 0.242820360467419 0.079318254340253 $0.00 $0.00
2 0 2 0 0.32665405070468 0.511685482273718 0.167144135471488 $0.00 $0.00
2 0 3 0 0.32665405070468 0.172311348781937 0.0562862000620066 $0.00 $0.00
2 0 5 0 0.32665405070468 0.061695385224797 0.0201530474934656 $0.00 $0.00
2 0 10 0 0.32665405070468 0.0114874232521291 0.00375241333746711 $0.00 $0.00
3 1 1 1 0.17499324144894 0.242820360467419 0.0424919219679937 $0.04 $0.06
3 1 2 2 0.17499324144894 0.511685482273718 0.089541501145442 $0.18 $0.23
3 1 3 3 0.17499324144894 0.172311348781937 0.03015332146179 $0.09 $0.12
3 1 5 5 0.17499324144894 0.061695385224797 0.0107962754429283 $0.05 $0.07
3 1 10 10 0.17499324144894 0.0114874232521291 0.002010221430786 $0.02 $0.03
4 3 1 3 0.05219096674793 0.242820360467419 0.0126730293588755 $0.04 $0.05
4 3 2 6 0.05219096674793 0.511685482273718 0.0267053599907461 $0.16 $0.21
4 3 3 9 0.05219096674793 0.172311348781937 0.00899309587456904 $0.08 $0.11
4 3 5 15 0.05219096674793 0.061695385224797 0.00321994179876811 $0.05 $0.06
4 3 10 30 0.05219096674793 0.0114874232521291 0.000599539724971269 $0.02 $0.02
5 15 1 15 0.00863850484104 0.242820360467419 0.00209760485940088 $0.03 $0.04
5 15 2 30 0.00863850484104 0.511685482273718 0.0044201975157114 $0.13 $0.17
5 15 3 45 0.00863850484104 0.172311348781937 0.00148851242061889 $0.07 $0.09
5 15 5 75 0.00863850484104 0.061695385224797 0.000532955883934236 $0.04 $0.05
5 15 10 150 0.00863850484104 0.0114874232521291 0.000099234161374593 $0.01 $0.02
6 125 1 125 0.00073207668144 0.242820360467419 0.000177763123677053 $0.02 $0.03
6 125 2 250 0.00073207668144 0.511685482273718 0.000374593009803969 $0.09 $0.12
6 125 3 375 0.00073207668144 0.172311348781937 0.000126145120390731 $0.05 $0.06
6 125 5 625 0.00073207668144 0.061695385224797 4.51657528755318E-05 $0.03 $0.04
6 125 10 1250 0.00073207668144 0.0114874232521291 8.40967469271539E-06 $0.01 $0.01
7 4000 1 4000 0.00002440255605 0.242820360467419 5.9254374563874E-06 $0.02 $0.03
7 4000 2 8000 0.00002440255605 0.511685482273718 1.24864336611557E-05 $0.10 $0.13
7 4000 3 12000 0.00002440255605 0.172311348781937 4.20483734670232E-06 $0.05 $0.07
7 4000 5 20000 0.00002440255605 0.061695385224797 1.50552509597445E-06 $0.03 $0.04
7 4000 10 40000 0.00002440255605 0.0114874232521291 2.80322489780154E-07 $0.01 $0.01
Return per 2$ bet: $1.43 $1.87
Return per 1$ bet: $0.72 $0.93
House Edge:28.260% 6.739%


Provided the approach above is correct, here are the overall results for all of the games offered:
Probability of a Win Probability of a Jackpot Win (1:n) House Edge House Edge w/ Multiplier HE after 30% Promo (no Multiplier) HE after 30% Promo w/ Multiplier
1 Spot 25.00% 4.00 50.00% 44.84% 35.00% 28.29%
2 Spot 6.01% 16.67 39.87% 33.67% 21.84% 13.77%
3 Spot 15.26% 72.07 37.56% 31.11% 18.83% 10.45%
4 Spot 25.89% 326.44 35.67% 30.32% 16.37% 9.41%
5 Spot 9.67% 1550.57 36.86% 30.34% 17.92% 9.44%
6 Spot 16.16% 7752.84 36.09% 29.49% 16.91% 8.33%
7 Spot 23.66% 40979.31 34.97% 28.26% 15.47% 6.74%
8 Spot 10.23% 230114.64 35.28% 28.60% 15.86% 7.17%
9 Spot 15.31% 1380687.64 37.16% 30.68% 18.31% 9.88%
10 Spot 5.90% 8911710.79 36.33% 29.76% 17.23% 8.68%
beachbumbabs
Administrator
beachbumbabs
Joined: May 21, 2013
  • Threads: 99
  • Posts: 14232
October 27th, 2013 at 9:02:52 AM permalink
I am not a mathematician, so the value of my feedback is limited, but it appears to me your approach is logical and correct. On the base game, your return goes from .65...to .845... with the 30% bonus, not counting the wheel multipliers. So I think you're in the ballpark of knowing your risk. So this moves their keno from a complete sucker bet into the range of a return comparable to slots. Still not a great bet, let alone at a +ev.
If the House lost every hand, they wouldn't deal the game.
craigGA
craigGA
Joined: Nov 29, 2009
  • Threads: 5
  • Posts: 17
November 7th, 2013 at 9:35:16 AM permalink
Thanks for the reply beachbumbabs! I am not a mathematician either...so I really value any feedback I can get.

  • Jump to: