I have a casino rental company that is outfitted with all kinds of standard games. I'm looking to create a new version of the "Big 6" or "Horse Racing" wheel games you may see from time to time.

Basically, the theory is I would have a wheel like this



split into 32 segments with 7 different colors representing different horses. You can see they are unevenly represented, to represent horses with different odds.

The idea is that players bet on a Horse to win and the operator spins the wheel. Whichever Horse is on the outside will move ahead 3 spaces on the game board. The middle horse will then move 2 spaces and the inner horse will move 1. Then the operator will repeat. Whichever Horse crosses the finish line (7 spaces away) first will win, and we will pay out.

My math is basic enough to figure out EV of each color horse for how many spaces they are likely to move per spin, but I can't figure out the math for what percentage of time the Red horse, or Blue horse, for example, will win the race.

Keeping in mind the complicated scenario of if the Orange horse is at position 6 and Blue is at position 5 and they move 1 and 2 spaces respectively, Blue will win because his move is resolved first.

Any help or insight would be greatly appreciated. Thanks

I think the Probability Forum is a better fit for this post. Moved from SMP.

Can I ask a couple of questions? And I apologize in advance if these are stupid questions but this is the first time I have heard of this game.

1. In your example above, why does Blue get "resolved first" (and therefore win that race)? What is the rule governing the order in which "moves" occur?

2. Are the three wheels "independent", meaning that the three winning colors of each spin are truly random and separate from which color wins in the other "wheels" of that spin? (I can imagine a giant fixed 3-wheel in which the colors of the three constituent wheels all move together. This would be the extreme opposite case of total dependence.)

No stupid questions. This game basically doesn't exist. I'm creating it, so my rules, in a sense

So movement resolution would be top down. Outside moves first, then middle, then inner.

No the wheels are not independent. Think of it as one slab of wood that gets spun around a hub in the middle. At the top there will be a little pointer (think Price is Right wheel) that points downward to indicate the winning "segment"

it's like that wheel for Wheel of Fortune show, if I got it correctly. so, for the image you posted, if the wheel stops at 12 o'clock, green would move 3, orange 2 and blue 1 in that priority. so if green moves to total 7 or more with this move and it's the first there, it wins no matter if orange would get at 7 also in this move? then if it stops right of that, green moves 3, grey 2 and blue 1.

this could be simulated with monte carlo relatively easy, 32 possible scenarios all with equal probability of happening and each moves the horses by some number of steps. not really sure if this is easy to solve analitically, I guess the minimum number of spins to win the race is 3 and by then you have 32*32*32 possible combinations of moving the horses.

Just to be clear on this last point, are you saying that the three wheels (outer, middle, inner) do not spin freely on their own? That they are "fixed" in position for each and every spin?

Alternatively, you may be saying that the three wheels move together on each spin but they their initial "configuration" (how the three wheels are initially "aligned") can be altered/adjusted/randomized.

If it is the first case, then to determine the underlying probability of each color winning the race I think would require the exact tri-wheel configuration.

Yes, correct.

Glad to hear it seems "easy", but I don't have the means to do this math lol

Yes, it is your first case.

I tried to draw it in a 'test' case where it's easy to count how may segments each color takes up. In some spots it's less clear because of my lines but you could round up or down. Does this help?

Ideally, I'd like some simulation where I could plug in how many segments each color takes up on each section of the wheel and see how probabilities change horse by horse. Because I don't have the exact configuration designed yet, and may want some horses more or less likely to win etc.

neither do I Monte Carlo method has more to do with programming than math. if you know a little bit of programming it's doable. I doubt you would get far with this on paper.

Yes, in the case of a fixed wheel, it seems like the resulting color odds are not easily found via pure math but are via simulation.

I think that you want all of the three circles to "line up" in three concentric 32/32/32 circles like slices of pizza. That is, I don't think it would be feasible or desirable to have any of the tiny segments "overlap" any of the associated segments of the other two wheels.

So if you want to find the true odds you'd simply need to specify the color outcomes of the 32 "slices" (outer, middle, inner). Then simulations could be performed to derive the underlying probability of each color winning the race.

For example:

1. O,B,R
2. O,G,W
3. Y,G,B
.
.
.
32. W,P,R

Feel free to post an example of your tri-wheel and I am confident that people in this forum would be happy to perform the simulations.

Since it only takes a few minutes, I decided to code up the simulation based upon my interpretation of the tri-wheel configuration in OP.

Here are the results for 1,000,000 games.

Color	Number of Wins
Red	386,509
Orange	287,639
Blue	113,279
Green	76,612
Purple	69,011
Yellow	40,720
White/Gray	26,230
.
TOTAL	1,000,000

Of course, the probability of each color winning is simply the number of wins divided by 1,000,000.

GREAT thanks! That gives me a good starting point.

So you could run any # of sims if I just give you # of segments for each color in each section?

The exact tri-wheel configuration would need to be specified. The three winning colors in each of the 32 "spokes" of the big wheel (outer, middle, inner).

See post #9 above for a sketch of what I mean. In case it is not clear, O would be Orange, B would be Blue, R would be Red, etc.

Right I guess you would need exact config to determine exact positions of the horses along the track. Ok let me whip up a few configs and get back to you. Thanks

Your results simulations are very helpful. I will likely end up using a very similar configuration and it's nice to know the "fastest" horse is roughly ~2.5: 1 and the "slowest" horse is ~40:1. Right away I can see this will work well for me to offer players 2:1 for betting on Red, for example, and ~30:1 for betting on the longshot gray.

Ok let's try this one, similar distribution but I'd like to see how the numbers change



Thanks

Here are the simulation results for 1,000,000 games using the 2nd wheel posted just above.

Color	Number of Wins
Red	394,279
Green	257,909
Yellow	111,515
Orange	106,363
Blue	80,728
White/Gray	24,658
Purple	24,548
.
TOTAL	1,000,000

Of course, the probability of each color winning is simply the number of wins divided by 1,000,000.

Sweet. Very similar. I don't suppose your data would havr the simulated "end positions" of each horse after the race? For example, purple only wins roughly every 40 times but I'm curious how many times he makes it to position 5 or 6 before the game ends?

Here are detailed simulation results for 1,000,000 games using the 2nd wheel posted just above, where I now show how many "points" each color had at the respective end positions. (Note that they are slightly different from the results posted earlier for the 2nd wheel due to randomness.)

Color	Zero	One	Two	Three	Four	Five	Six	Seven
Red	48,326	37,927	70,571	116,723	90,340	106,960	134,549	394,604
Green	101,237	104,345	85,111	137,887	115,542	91,606	106,604	257,668
Yellow	218,601	123,303	102,155	187,939	94,479	74,473	87,427	111,623
Orange	120,223	82,049	216,399	165,526	123,482	116,425	69,670	106,226
Blue	214,470	141,765	122,707	188,430	104,603	73,957	73,234	80,834
White	349,995	172,805	149,756	156,008	68,866	45,383	32,594	24,593
Purple	248,732	134,640	217,979	161,194	94,941	79,876	38,186	24,452

Of course, the probability of each color winning is simply the number of wins (times ending with 7 points) divided by 1,000,000.

I've made a script which you can run online here https://repl.it/repls/KookyJuicyActionscript
click console and run and wait a couple of moments. it's only for 10000 races, this online compiler can't handle much more, I can run 1 million on my PC in a minute so if you want better accuracy, download Python and change "Nraces" variable.

for steps when it loses, I programmed it so it can't have 7 steps because the first one that gets to 7 or more wins and it's over at that moment. you can change wheel configuration, I hope it's clear how it works. also, you can change number of steps required to win from 7 to something else.

Sick, thanks guys!

One last question, is there a way to see average # of spins to resolve the race?

possible number of spins: [3, 4, 5, 6, 7]
percentages: [26.2897, 37.7285, 27.1936, 8.1887, 0.5995]

average is 4.19 spins.

My man.

So my next step is designing a configuration such that "true odds" for each color to win is something like this:

Color 1: ~31%
Color 2: ~23.00%
Color 3: ~18.50%
Color 4: ~15.00%
Color 5: ~7.00%
Color 6: ~3.50%
Color 7: ~2.00%

So that I can offer player odds of 2:1, 3:1, 4:1, 5:1, 10:1, 25:1, 40:1 respectively.

We're in the ballpark with these configurations and I've tried some new ones but haven't been able to get too close. It's tough when you need to slightly raise one and decrease another, you move one color and several outcomes could change - this brings me back to my University days

If anyone wants to try and have a fiddle be my guest!

Edit: My only constraints are that colors must have a minimum of 2 touching segments on outside, 3 in middle and 4 inside, because of total area restrictions that I want to be able to add images. Potentially these constraints could change if there's no way we can get close to those numbers.

Had to alter my desired odds a bit but this config turned out pretty nicely:

Color	True percentage	True odds	Odds to offer
red	29.26%	2.4	2
green	23.00%	3.3	3
yellow	18.60%	4.4	4
orange	13.32%	6.5	5
blue	8.39%	10.9	10
purple	4.44%	21.5	20
grey	2.99%	32.4	30

only suckers are going to bet on orange