roulette math help

[toastblows]

I played roulette recently. I know that a live wheel can be biased unlike the software wheels which are all RNG driven. This was a live wheel.

I played 4.5 hours, average 25 spins an hour i would guess.

Bet was 2x on 1/3 of the board, 2x on a 2nd 1/3 and 1x on 1/3 and nothing on 0 or 00.

My conslusion was that the 1x 1/3 was being hit 1 out of 5 times instead of 1 of approx 3. and 0 and 00 were not being hit 1 of 19 times. 0 and 00 hit 5 times in 4.5 hours.

So based on my betting style, and a profit of 12 units, can anyone tell me what my risk of ruin was, and if i had any advantage based on a 12 unit gain betting that way...if indeed there was a bias? (obviously 4.5 hours at a wheel is not enough analysis to tell if its bias'ed, so it most likely was luck...just wondering though)

 

[Guynoire]

As you guessed inconclusive results

Yeah, you’re basically right it’s not enough data to determine an advantage. Also it would have been nice to have some real numbers instead of just guesses because I think we all have selective memories, here’s the analysis anyway.

Inference of Data:

Number of Spins 114:
Number of wins: 85
Number of losses one 1/3rd of table: 24
Number of 0’s losses: 5

This was just algebra using the fact that you won 12 bets and lost on the 0’s spin 5 times, it must have landed on the losing one third of the table 24 times if the total number of spins is 114.

From your data I’m going to change your system a little because it is not the optimum system for winning; putting money on the losing 3rd only gives the house an advantage because you lose more on the 0’s spins. You might have a reason to bet this way but I’m first going to look at what would happen with the strongest system, betting only on the two winning thirds.

Transform the data:

Wins 85
Losses 29

Using a binomial test the probability of results at least this extreme occurring is about 0.6% so there might be reason to believe the wheel is biased, but the question still remains as to whether it is biased enough.

95% confidence interval of player advantage: (-.3% to 47.9%)

Even with the strongest system imaginable and questionable data it is still inconclusive even at the 5% level whether you can attain an advantage from this wheel because the confidence interval goes into the negatives. If you really want to determine if the wheel is biased you’re going to need thousands of spins of data. It would be a lot better if you could record exactly which numbers hit as you might find that it is not exactly one third that is favored.

 

[toastblows]

Thanks for the analysis. My hunch was that i was just "lucky". And when I think back i selectively pick out 5 or 6 numbers that kept hitting in my mind but who knows. I do know that based on my 4.5 hours, one of the 1/3s was hitting more than the other 2 which is why it was always a 2x bet.

I did other variations with 2x on that 1/3, 1x on the 1/3 i felt 2nd most...but it was too inconsistant between the 1x 1/3 and the no bet 1/3 in that system. So my logic was that 2x, 2x, 1x winning on the 1x would cover the loss of one 2x...rather than a straight loss of 2x and 1x (so losing 2x each time instead of 3x each time). Of course in that system you cant have a 0/00 or you get screwed worse over. I was lucky to not have a 0/00 for over 1 hr toward the end and a lot of hits on 2x 1/3s....pure luck is my conclusion over 114 spins you calculated.

Thanks again
:cool2:

 

[Brock Windsor]

I am new to biased wheels so could use some help. Can one number be biased as opposed to a section of numbers? I recently tracked 622 spins of which the number 4 appeared 33 times! Its neighbours appeared only 11 and 12 times. Another section of the wheel (00,27,10) appeared 25-19-23 times respectively. I had thought if the wheel showed a bias there should be a section of numbers hitting such as the latter as opposed to a single number appearing many times. My plan right now is to bet the minimum $1 on number 4 until the data normalizes...should the advantage persist through 1000 spins I will be looking to bet some fraction of kelly (maybe 1/10th?) Any thoughts on this? I won't bet more than the minimum until I have at least 1000 spins and a 95% CI the wheel is biased to a player advantage. If CI drops below 95% after 1000 spins I would bet the minimum or quit altogether.
BW

 

[Sonny]

Can one number be biased as opposed to a section of numbers?

Sure. If that particular number has damaged or improperly installed frets, or maybe the pocket is damaged or too deep for some reason.

I won't bet more than the minimum until I have at least 1000 spins and a 95% CI the wheel is biased to a player advantage.

Unless the bias is very large it may take more than 1000 spins to get reliable results. Usually somewhere between 2,000 and 8,000 spins will give you a good indicator depending on the amount of bias. Also, be aware that the bias may change as time goes by. If the bias is caused by something small like dirt buildup or a slight tilt then the conditions can quickly change. Certain conditions may disappear while new conditions begin to form. A number that is biased today may not be biased next week. This is especially true if the casino services their wheels regularly and/or runs weekly tests on them. If you discover a bias, they might too.

Beating the Wheel by Russell Barnhart is a decent introduction to roulette biases. It covers how to tell if a wheel is biased and to what degree. It doesn’t directly tell how to detect a bias on certain numbers but it will give you the gist of it.

-Sonny-

 

[Brock Windsor]

Thanks Sonny. I'll stick to $1 betting and post my results at 2000 spins and try to get a consensus on how to bet my edge if one still exists....again this is my first attempt at advantage roulette so I only want to stick my toes in the water before I dive in.
BW

 

[Guynoire]

Wow, 33 times in 622 spins! That's as unlikely as rolling the dice 50 times before sevening out. More improbable things have happened in a casino but I would suspect a bias.

 

[Sonny]

Beating the Wheel by Russell Barnhart is a decent introduction to roulette biases. It covers how to tell if a wheel is biased and to what degree. It doesn’t directly tell how to detect a bias on certain numbers but it will give you the gist of it.

My mistake. The book actually does cover a method for determining the bias of a particular number or series of numbers. Based on your results above, the number 4 has over a 95% CI of being biased. It comes up about twice every 38 spins which gives you over an 89% advantage. The variance on a single straight up bet is pretty high (33.21) but you can safely play this number while you continue to clock the wheel.

The numbers 00, 27 and 10 do not have even an 80% CI. The book recommends at least an 80% CI before playing a particular number, although the exact percentage is up to the player.

-Sonny-

 

[Kasi]

The book actually does cover a method for determining the bias of a particular number or series of numbers. Based on your results above, the number 4 has over a 95% CI of being biased. /QUOTE]

I don't know about the book or anything but, help me out here, in answering what are the chances of any number appearing exactly 33 times in 622 spins in an unbiased 0,00 wheel?

I get like 1 in 312?. Is that what you/anyone gets? Is it the right question to even ask? It just seems a little dangerous to me to log 622 spins, determine what number hit the most, and then potentially conclude it's probably biased and start betting on it for something that seems like it's going to happen randomly, with one number or another, every 1000 spins or so anyway.

If a number hits 5 out of 10 spins, I think a more rare event, would that be enough to conclude the number is biased and to begin betting on it?

Put another way, is this really only a 2-3 standard deviation event on a fair wheel when considering the chances of it happening to any of the 38 numbers?

I guess you got me wondering what this guy's CI index is based on. Are high tides involved?

Much more logical to conclude, as time goes by, after you've lost your money betting on that 4, that the wheel was actually biased for 4 back then, but, if and when the bias has apparently disappeared a few thousand spins later and everything is pretty normal, as can often happen as you say, and even maybe switch to a different number, that they must have fixed the fret or, more likely, that speck of dust blew out of the 4 and lodged in the next obviously biased number. Anything is better, from the author's point of view, than concluding the wheel was actually never biased at all.

That way, after you've lost your money, you were merely incredibly unlucky rather than stupid.

And, of course, should you win, it was pre-ordained and solely due to your ability to recognize a biased wheel. Really, when you think about, doesn't even qualify as gambling what with that 89% advantage. If you believe it, Brock, bet a lot more than a dollar before a speck of dust blows away that rare 89% advantage. Keep us posted on results of spins.

Just a starting point for discussion lol. I've eaten crow before lol.

 

[Guynoire]

I don't know about the book or anything but, help me out here, in answering what are the chances of any number appearing exactly 33 times in 622 spins in an unbiased 0,00 wheel?

The chances of landing exactly 33 out of the 622 possible is very very small, but this number really doesn't mean anything. The probability of getting any exact number out of 622 is pretty small. The real number you want is the probability of getting results at least that extreme ie.>=33

The distribution is just binomial so I used my calculator to add up all the possibilities. The probability of results at least this extreme occurring due to random chance is about 1.5*10^(-4) or about 3 out of 20,000.

If you construct a 95% confidence interval of your projected advantage the size of this interval is still pretty big because of the relatively small number of counts but I get a range of (27%-154%) with a middle of 90% just like Sonny's numbers.

 

[Kasi]

The chances of landing exactly 33 out of the 622 possible is very very small, but this number really doesn't mean anything. The probability of getting any exact number out of 622 is pretty small. The real number you want is the probability of getting results at least that extreme ie.>=33

What I said was meant to apply to the chances of getting any 1 of the 38 numbers 33 times in 622 spins. The chances of getting a particular number, say 4 in this case, is what I said times 38. Or 1 in 11,842.

FWIW, for a particular number, I get about 1 in 14059 for occurrring more than 33 times in 622 spins. I'm using a combin function because I'm not sure the binomial thing works that well for very low probabilities. Even a 1 in 14000 probability falls far short in my mind to conclude something is wrong.

Basically, my point is, I think the question to ask is what are the chances of any number, rather than a particular number, occurring this often.

Assuming for a second that what I said is correct, (big assumption I realize and I'm not denying it), that any of the 38 numbers would occur this often in that many spins, that is .32% of the time, should one accept this, is that enough for you to assume the wheel is biased?

Personally, just because a number actually happened to hit exactly 33 times in 622 spins, and knowing (or at least believing for the moment lol) that this can be expected to happen for any of the 38 numbers once every 312 times that 622 consecutive spins occur, isn't enough for me to conclude things are biased.

Even by your own numbers that seem to be 3 out of 20000, or 1 in 6666, is a 1 in 6666 event enough for you to conclude a bias exists? I guess apparently it is for you but getting 3 BJ's in a row is more rare than that and it doesn't make me think the game is biased in my favor when it happens.

Anyway, I wouldn't mortgage my house to start betting that apparent Monday-morning quarterbacking 89% advantage as if I believed it. If I did I would. How would you start betting Brock's sure-thing wheel beginning with the 623rd spin with that 89% advantage on 4? Retirement could be but a few hours away.

 

[Sonny]

FWIW, for a particular number, I get about 1 in 14059 for occurrring more than 33 times in 622 spins.

I get the same answer using the binomial method. Yay!

Basically, my point is, I think the question to ask is what are the chances of any number, rather than a particular number, occurring this often.

For that kind of analysis you can use a Chi Square test. It will look at the results of every number on the wheel and tell you how likely it is to be biased. It won’t tell you which number or section is biased, only how likely the overall results of the wheel are. This is a good place to start. If something is “funny” you will know right away. You won’t know exactly what is causing the “funniness” or where it is happening, but it will let you know whether you should keep looking or not.

I guess you got me wondering what this guy's CI index is based on. Are high tides involved?

His CI numbers are based on the degree of volatility and the number of trials. If the number of trials is small then the confidence level will be small unless the results are extreeeeeemely unlikely. In the case of roulette you will often be looking for a very small bias over thousands of trials so the formula seems appropriate. Basically, he’s looking at the Standard Error of the test. Every sample is going to be unreliable to some degree. There’s no way to escape that fact. You will never get a CI of 100% so you have to decide how much confidence you personally need before you put your money on the table. A very good level is 95% or better, but it can often take several months or a year to get enough roulette data for that kind of certainty. The author suggest 80% as a good starting point so that you can take advantage of a potential bias as you continue clocking the wheel.

Much more logical to conclude, as time goes by, after you've lost your money betting on that 4, that the wheel was actually biased for 4 back then, but, if and when the bias has apparently disappeared a few thousand spins later and everything is pretty normal, as can often happen as you say, and even maybe switch to a different number, that they must have fixed the fret or, more likely, that speck of dust blew out of the 4 and lodged in the next obviously biased number. Anything is better, from the author's point of view, than concluding the wheel was actually never biased at all.

I won’t disagree with you there. To be honest, parts of the book did make me shudder a little bit. Some of the math looked a little different to me (not necessarily wrong, just different than I would have done it). The book was obviously intended for serious players but there are a few paragraphs thrown in for the gamblers too. There are a few “Don’t ever start betting until you’re pretty certain of a bias…but if you really want to…” parts that still make me cringe. Despite this, I still believe that the overall information and concepts in the book are good to know.

-Sonny-

 

[Guynoire]

I see what you're saying you're talking about the probability that the highest of all the numbers is greater than or equal to 33. For this I got a probability of about 0.6%

For me, this number is still strong enough to suspect a bias assuming all the data is accurate and that I had never observed this wheel before. The lower bound of the 99% confidence interval still gives you a 7.5% advantage. I think this serves as a good conservative estimate until you get more data. I would start betting using fractional kelly with this 7.5% advantage as the estimate.

 

[Kasi]

I get the same answer using the binomial method. Yay!

No one more surprised than I lol. (Not you getting it right, me lol.)

Don't mean to beat this to death but, just so I think I might have a clue what I'm doing, would you, or Guynoire, say getting a 4 33 times in 622 spins is less than a 3 Stan Dev event? (Sometimes I get close to 4 and sometimes 2.8 or so lol.) I am with you on your variance number.
Never took any classes in prob or statistics so it's always a laborious process for me. Anyway, if less than 3, what is the big deal? I guess my personal threshold for concluding something is fixed, biased, cheating, whatever, begins to kick in around 4 SD. Less than that, that's the way the ball bounces. Do you have a threshold?

Maybe a Chi-square test is a better way of determining this? Haven't done one - didn't think of it.

Anyway, I hope Brock posts results for the next 1000 spins or so to see what they may bring.

And to see whether Guynoire has made some "play" money playing that 4 :grin:

 

[Guynoire]

Gonna try to settle the math once and for all

We all know that the binomial distribution approximates to normal with a mean of the expected amount and a standard dev of the (p*(1-p)/n)^1/2 where p is the probability and n is the number of trials. If you were to test this with the assumed percentage of 1/38 you would get about 4 standard devs away from the expected amount.

However, as you asserted this is an unfair test because Brock Windsor didn't tell us the results of any random number he told us the amount of the highest number.

To compute this distribution: Pr(X highest<=x)= (Pr(X<=x))^38
ie. The probability the highest is less than a number is the probability that they're all less.

differentiating we get
f(x) highest=38*(binomcdf(X))^37*binompdf(x)
you could approximate binomcdf with normal cdf if you wanted to.

normal cdf= .5*[1+erf{(x-E(x))/(sqrt2*std.dev)}]
f(x) highest=[normal cdf(x)]^37* normal pdf(x)

To tell the truth I really don't know what the distribution of the highest number looks like graphically but I doubt it is normal so its standard deviation wouldn't have relative signifigance.

It's pretty easy to calculate the p value P(X>=33)=1-[binom cfd(32)]^38

For this number I get 0.589% I don't think it's a suitable time to use the normal approximation because the difference between the 2 will be compounded by a power of 38.

I hope this math is easy enough to follow and I haven't made any major mistakes. Results like this should happen just by random chance a little more than once in 200 times. Beyond this the question stops being mathematical, unless you believe in Bayesian statistics and want to put an arbirtrary priori function on p to calculate an expected value with an appropriate loss function. But I don't believe in Bayesian statistics.

 

[Kasi]

I hope this math is easy enough to follow and I haven't made any major mistakes. Results like this should happen just by random chance a little more than once in 200 times.

Thanks Guynoire. Much appreciated.