Roulette Wheel Question

wayfarer

Here's one I just started thinking about:

You're playing roulette, and can only bet on black or red. Instead of having the usual casino odds of 18/37 chance of hitting, the 0 counts as whatever colour you are betting on, so you're getting odds of 19/37 for a 1:1 return ie. so long as you don't go broke, you'll eventually make money.

You start with E100 and must place 500 bets.

What is the correct amount to bet to get the maximum (average?*) return? if (a) If you must bet the same amount for each of the 500 bets (b) If you can bet any amount at any point.

How does the amount that you bet relate to the probability that you will win; the total amount of money that you have; the amount of bets that you have left etc.

If you bet too much you risk going broke, if you bet too little you are not maximizing your profit.

There might be a formula for this but I haven't a clue what it is

*what I mean here is if you repeat this amount for 500 bets over and over, your profit will average will be more than than the average you will earn if you bet any other amount. Im as clear as mud I know

Pherekydes

There are 37 slots on a roulette wheel: 18 of each colour and one green. If you bet on red your expectation is -1/37.

To maximise your profit, bet nothing on each of the 500 bets.

Smurfpiss

Slow coach wrote:

There are 37 slots on a roulette wheel: 18 of each colour and one green. If you bet on red your expectation is -1/37.

To maximise your profit, bet nothing on each of the 500 bets.

That doesn't maximize your profit, it minimizes your loss

wayfarer

Sorry, got the number of slots wrong. Duly edited. I know your odds are usually 18/37 but for this question the gambler has the edge instead of the house. Why, you ask? Coz its CRAZZY CASINO!!!

Pherekydes

Smurfpiss wrote:

That doesn't maximize your profit, it minimizes your loss

The same thing...

To maximise your profit, I think you should place 500 even sized bets, but I'm open to correction.

Capt'n Midnight

€1 x 500

You'd assume that on average you would win back what you put on . So maybe sometimes you will do better, sometimes you would do worse but on average you should break even.

Actually not because a roulette wheel has no memory. If you loose €100 during your first plays, it would be as if you were starting with €400 instead of €500, and the roulette wheel would not be obliged to give you your money back.

Playing double or quits guarantees you will double your bet, but ONLY if you have an infinite amount of money to gamble AND the house doesn't take any % of anything.

If you had another €11 you could place 8 bets
1
2
4
8
16
32
64
128
256
But spending €511 to chase €1 you lost on the first bet is kinda daft, even if you win you could be back below your starting point on the next bet.


Overall you have to be lucky with your first few bets, and profit take immediately because the wheel has no memory and won't help you to win back any loss.

Or you would just bet 1c at a time as the probability is that you would end up a few c ahead at some point. - I'm not too sure on this, it's rather like the way you could get free energy by separating slower and faster molecules one by one - possible in theory but well nigh impossible.

red_fox

Minimise your loss, bet as little as possible, ideally nothing.

Maximise your profit (accepting hugh risk, not assumed in basic game theory) bet everything everytime, and there's a 1 in 2^500 chance of leaving with a very large sum.

For the greedy who aren't so foolhardy then they must decide an acceptable risk threshold and calculate the amount of money to bet each time (same size bets seem to make more sense, but a certain percentage of your total could also be investigated although that has a greater potential for loss than gain) so you have an acceptable risk of going bust. It would only be a few euro. Maximum risk gives maximum potential profit.

Of course then you could also set an amount that you can't accept losing, basically refusing to bet that amount, if you try to avoid any loss as in game theory then you simply neve bet, even if the odds are in your favor.

Noelie

your initial thinking is wrong, you should be using the zero as a push, so if you land on zero you neither win nor lose, in terms of roulette the zero is what give the house its advantage, if you use the zero as a wild color, you take the advantage that the house had. You now have the houses 2.7% advantage.


I would suggest betting nothing for the first ten-20 spins then pick a color and bet on that, each time you lose double up plus 1, so you would be betting 1,3,7,15,31 etc, the bets raise quicker but you also make back more than your initial 1 unit bet, e.g win with a bet of 31, you get 31 units back(not including your stake) your previous bets would have cost 26 so you have won 5. once you win change color and continue in this manner. the longer your winning streak gose the high your potential winning could be.

Pherekydes

Noelie wrote:

I would suggest betting nothing for the first ten-20 spins...

The point being? A roulette wheel has no memory, so observing it for x number of spins will have no effect on later spins.

Noelie

I know it has no memory, but its nice to watch going around

yaledo

I'm assuming that if you are down to less than your chosen stake, that you can bet your remaining cash
i.e. lets say you choose 97 as your stake and you lose on the first go... are you allowed bet with 3 the next time (and then 6 and 12 until you make it back up to 97)?

Given that assumption, here are the results from 1,000,000 tests for each number between 0 and 100 - the 'times broke' is how many times out of the 1m attempts you went broke.

Bet Size | Average | times broke
Payoff
1 | 113 | 0
2 | 126 | 3284
3 | 139 | 25480
4 | 150 | 67116
5 | 158 | 108947
6 | 166 | 142670
7 | 174 | 169520
8 | 177 | 201291
9 | 183 | 219122
10 | 179 | 256694
11 | 192 | 256246
12 | 190 | 278340
13 | 186 | 300903
14 | 198 | 300479
15 | 189 | 324031
16 | 198 | 324370
17 | 183 | 349522
18 | 192 | 349396
19 | 201 | 349412
20 | 178 | 375640
21 | 184 | 376184
22 | 192 | 376110
23 | 200 | 375990
24 | 207 | 376450
25 | 169 | 404821
26 | 176 | 404387
27 | 182 | 404445
28 | 188 | 404667
29 | 195 | 404078
30 | 200 | 404904
31 | 208 | 404039
32 | 213 | 404627
33 | 218 | 404960
34 | 158 | 434473
35 | 161 | 434828
36 | 167 | 434562
37 | 171 | 434452
38 | 175 | 434492
39 | 178 | 434825
40 | 184 | 434525
41 | 187 | 434879
42 | 191 | 434621
43 | 195 | 434825
44 | 201 | 434552
45 | 205 | 434545
46 | 209 | 434421
47 | 213 | 434567
48 | 216 | 434981
49 | 221 | 434811
50 | 120 | 466110
51 | 121 | 466587
52 | 123 | 466417
53 | 126 | 466375
54 | 129 | 466369
55 | 130 | 466482
56 | 133 | 466284
57 | 135 | 466506
58 | 137 | 466439
59 | 141 | 466175
60 | 142 | 466296
61 | 145 | 466340
62 | 146 | 466512
63 | 148 | 466504
64 | 151 | 466418
65 | 152 | 466583
66 | 156 | 466318
67 | 156 | 466687
68 | 159 | 466607
69 | 163 | 466280
70 | 165 | 466241
71 | 167 | 466439
72 | 168 | 466584
73 | 171 | 466400
74 | 172 | 466680
75 | 175 | 466591
76 | 176 | 466599
77 | 182 | 466081
78 | 184 | 466128
79 | 184 | 466465
80 | 188 | 466208
81 | 190 | 466236
82 | 190 | 466638
83 | 192 | 466563
84 | 195 | 466548
85 | 197 | 466523
86 | 200 | 466428
87 | 204 | 466397
88 | 206 | 466187
89 | 210 | 466130
90 | 209 | 466671
91 | 213 | 466231
92 | 213 | 466484
93 | 217 | 466298
94 | 219 | 466233
95 | 220 | 466572
96 | 224 | 466091
97 | 225 | 466480
98 | 226 | 466456
99 | 230 | 466429
100 | 0 | 500000

red_fox

Could you change that for the 500 bets in the problem? Also if you wrap it in code tags it will keep the spacing and everything should line up nicely.

[*CODE] without the * [/code]:)

Mr. Flibble

wayfarer, no strategy is better than any other; whatever strategy you use (provided you actually bet some money once) will give you the same results in the long run.
One could come up with a formula to relate bet sizes to chances of going broke, but weather you chose a low or high varience method you will make the same amount in the long run.

Sorry, I can't come up with a formula. I'd imagine the chance of going broke would increase linearly related to bet sizes.

can anyone remember that variance formula - usually explained by a drunk stumbling along a road, where he takes random step to his left or right and will eventually hit the side of the road. did a quick google search but couldnt find it. that formula would be exactly what you are looking for if you had 0 edge in the roulette game. if you factored in your 2.7% winnings somehow you should have the answer.

edit: any and all of the above could be wrong - just rereading the 1st sentence - I think that this is right provided the same of money is bet with whatever strategy choosen.

Mr. Flibble

Ok, bit slow today. I've finally copped the obvious answer. As you have an edge then by betting the maximum possible will give you best advantage. So bet everything you have each time and you will win most on average. You might want quite a large bankroll to do this as its possible you will go through a fair few 100euro lumps!

ecksor

Mr. Flibble, I read the original question as assuming that the entire bankroll is E100. Everyone has a limited bankroll, so the question about what the proper bet size is a good one and you can't just bet all your money every time you have an edge or you will eventually go broke.

wayfarer wrote:

If you bet too much you risk going broke, if you bet too little you are not maximizing your profit.

There might be a formula for this but I haven't a clue what it is

The formula you're looking for is the Kelly Criterion.

http://en.wikipedia.org/wiki/Kelly_criterion

Mr. Flibble

ecksor wrote:

Mr. Flibble, I read the original question as assuming that the entire bankroll is E100. Everyone has a limited bankroll, so the question about what the proper bet size is a good one and you can't just bet all your money every time you have an edge or you will eventually go broke.

Yes, most likely you will go broke, but in the highly unlikly case that you make it to 500 spins this method will give you the highest return.

ecksor

Reread the original post. That can't be the answer that wayfarer was looking for.

wayfarer wrote:

If you bet too much you risk going broke, if you bet too little you are not maximizing your profit.

By that rule, your method is betting too much.

Mr. Flibble

wayfarer wrote:

If you bet too much you risk going broke, if you bet too little you are not maximizing your profit.

I read that as a statement not a rule

Yes you risk going broke with this method. I'm not arguing that the Kelly method isn't best if you wish not to go broke - but my method will give greatest returns if you repete the process over and over again.

I agree the Kelly Criterion is probably what wayfarer is looking for to answer (B), athough my method has a place:D

ecksor

my method will give greatest returns if you repete the process over and over again.

I'm not convinced about that. When you eventually win 500 in a row you'll have 100 * (19/37)^500 I think? How many E100 lumps do you expect to invest before you'll do that? EV wise, any scheme should be the same as any other since you have the same edge on every spin.

athough my method has a place

I can't tell if you're joking here. A method that expects to lose all of your money all but one time out of (19/37)^500 trials has a place?

Mr. Flibble

ecksor wrote:

I'm not convinced about that. When you eventually win 500 in a row you'll have 100 * (19/37)^500 I think? How many E100 lumps do you expect to invest before you'll do that? EV wise, any scheme should be the same as any other since you have the same edge on every spin.

Yes, EV wise every method is the same, but betting more makes more use of your +EV. Say you do the experiment a billion times long enough for any varience to die, doesn't logic dictate that more money will be made if more money is wagered?

If you say that you will actually get to wager less in a billion goes of 500 bets because you will often lose the $100 and not get to use the rest of your bets from the 500, and have to move onto the next go then this method probably isn't optimal. In this case it looks like Kellys' way is best.

ecksor wrote:

I can't tell if you're joking here. A method that expects to lose all of your money all but one time out of (19/37)^500 trials has a place?

I wasn't joking. I'm pretty sure that over an infinate amount of runs you will make more money if you wager more money. It has a place in the theory side of things, if not the real world.

ecksor

In the same way that 10*1.5 is greater than 1*1.5, then yes, you'll make more if you bet more. That doesn't make the system "better", it just means that you invested more money on it. If I go to someone asking advice on the best way of investing E100, I don't think I'm looking for the answer of "Don't start with E100, start with E1,000, you'll make much more money!"

I wasn't joking. I'm pretty sure that over an infinate amount of runs you will make more money if you wager more money. It has a place in the theory side of things, if not the real world.

Over an infinite amount of runs, you'll make an infinite amount of money if you don't go broke at any stage, so betting 1 euro per spin seems to match your system there in terms of "money made". I'm not being smart btw, you're simply not demonstrating that you'll make more money per euro invested using your method over the long run. Show me some calculations that say otherwise.

ecksor

Some numbers to illustrate what I'm getting at:

As a simplification, if I look at starting with E90 and only having 3 bets at it, then if I bet 90, then 180 if I double up, then 360 if i double up again, I end up E720 every (19/37)^3 times, which is approximately E97.49. So, I make E7.49.

If I bet E30 3 times, then I get an average return of E60*(19/37) which is E30.81 so my overall result is E92.43 so I make E2.43 which is less than I make in your scheme if I try it an infinite number of times.

So far, we are making much more by betting the maximum.

However, in the second scheme I risk E90 to get my average of E2.43 profit whereas in the first scheme I appear to risk a combination of E(90+((19/37)*180)+((19/37)^2 * 360)) = 90 + 92.43 + 93.6 = E276.03 to get my average of E7.49 profit.

2.43/90 = 0.27
7.49/276.03 = 0.27

So, for every euro I risk, I make 27c with both methods.

EDIT: Clearly wrong, that should be 2.7c per euro invested.

Mr. Flibble

I agree with all you have said. I'm just pointing out, albiet in a rather convoluted way, that to get best return when you have a max $100 to bet is to invest the whole $100.

If I go to someone asking advice on the best way of investing E100, I don't think I'm looking for the answer of "Don't start with E100, start with E1,000, you'll make much more money!"

I'm not saying this, I'm saying invest your whole $100, not invest only 50 cent.

ecksor

You said this:

Say you do the experiment a billion times long enough for any varience to die, doesn't logic dictate that more money will be made if more money is wagered?

I'm just pointing out, albiet in a rather convoluted way, that to get best return when you have a max $100 to bet is to invest the whole $100.

How is it the best return? Your scheme has the best possible outcome if everything goes right, but you've agreed that per euro invested your EV isn't altered. What exactly do you mean by "best return?