Is this roulette wheel biased

I have a record of 1363 consecutive spins from a single zero roulette wheel in Macau. The data is good as I collected it myself with the help of a friend.

The mean of the spins is 37 and the standard deviation is 5.99. The lowest number of outcomes for any number is 26 (#21), and the highest is 51 (#17).

There are 10 numbers that have occurred more than 40 times, which is around 10% or more above the mean.

The numbers that are 10%+ above the mean 28, 22, 9, 30, 3, 7, 14, 25, 13, 17.

Of interest to me is that these numbers sit in 2 sectors of the wheel that are opposite each other.

My question is, is this a biased wheel. Lets know your views.

The number of occurrences of 26 is above 2 SD below the mean and the number of occurrences of 17 is less than 3 SD above the mean. Given that there are 37 numbers on the wheel, it seems quite normal to me that some of them would deviate from the mean like this. Try getting something that deviates more than 3 SD from the mean!

I would also say that the sample size is to small. Maybe after 10,000 spins you would have a more clearer picture.
It's also worth noting that if the wheel was indeed biased, the casino would know about it before you do.

>I have a record of 1363 consecutive spins from a single zero roulette wheel ...
A sample size from which you can make absolutely no valid conclusions whatsoever.
>The data is good as I collected it
No observer should ever be certain of his observations. None!
>Of interest to me is that these numbers sit in 2 sectors of the wheel that are opposite each other.
Then how can the wheel be tilted? Bias would mean a tendency to have results in one particular sector over all others.

If you ENJOY gathering data and analyzing it, ... good for you.
If you think it will be profitable to do this at a roulette wheel, you are wasting your time and efforts.

Quote: Pando

I have a record of 1363 consecutive spins from a single zero roulette wheel in Macau. .

Your sample size is so small that its worthless.

Thanks for the replies guys, I will try and get 10,000 spins data

Pando, before you start on this time consuming endeavor, will you be able to be certain that it is the same wheel? That the casino did not change it out for service, or clean it, or fix it? Good luck.

I would not consider this project as anything but an exercise in mathematics. The wheel might be moved simply to vacuum the carpet or something and then repositioned. This would destroy the value of the data and yet would be an unknown event for the observer. If as a hobby you wish to record 10,000 actual wheel spins, thats fine. If you want to do it for profit, that is a waste of time.

Your life. Your decision.

I'm told there is a two decade old listing of 10,000 roulette spins available. There are certainly ways to obtain 10,000 random number samplings. Its your time. I'd suggest you simply make a few bets and enjoy the game instead of trying to prove some ever so slight bias in the wheel. Its probably non-existent anyway.

Can you give me the full table of how often EVERY number hit?

Hi Wizard.
Numbers and hits as follows in increasing order of hits;

21 (26 hits)
18 (27)
1 (27)
12 (29)
5 (29)
8 (29)
26 (30)
34 (31)
20 (32)
15 (32)
11 (33)
19 (33)
0 (33)
27 (34)
36 (35)
33 (36)
23 (36)
31 (37)
32 (37)
4 (38)
10 (38)
16 (40)
6 (40)
2 (40)
35 (40)
24 (40)
29 (40)
28 (41)
22 (42)
9 (43)
30 (43)
3 (43)
7 (44)
14 (44)
25 (45)
13 (45)
17 (51)

Thanks
Its something I enjoy and I will certainly make some bets while I am there

I have been at this wheel all hours of the day and have never seen it moved, maintained or adjusted. But of course that may have happened when I was not there, that I understand.

Its a Megastar air-wheel and I suspect these get less attention than wheels with a real croupier.

The wheel is obviously biased towards odd prime numbers between 3 and 17 that do not include the number of players on the field for a football or basketball team. If you notice those 4 numbers are amongst the 6 top hits!!! I may sell this system.....

Thank you. Those results have a chi-squared statistic of 36.295 with 36 degrees of freedom. The chances that a fair game would have results more skewed are 45.5%. So this is just about exactly the amount of variation you would expect to see in a fair game.

you can do the test yourself in excel. CHITEST( actual hits range, expected hits range) will give you the probability your results are random. Anything higher than 0.05 is generally considered random. You would need the CHITEST to be less than 0.05 to have any reason to believe you had found something fishy.

Quote: Pando

21 (26 hits)
18 (27)
1 (27)
12 (29)
5 (29)
8 (29)
26 (30)
....
....
9 (43)
30 (43)
3 (43)
7 (44)
14 (44)
25 (45)
13 (45)
17 (51)

How a non-math person analyzes this:
Take the fewest appearing numbers and the most appearing numbers: Are they in the same or different sectors of the wheel?
It would be impossible to have a bias so fine that it favored one number but suppressed its neighbor. The bias would be to a particular sector only. Any bias within a sector would be hidden until there were zillions of results in your data.

Quote: FleaStiff

How a non-math person analyzes this:
Take the fewest appearing numbers and the most appearing numbers: Are they in the same or different sectors of the wheel?
It would be impossible to have a bias so fine that it favored one number but suppressed its neighbor. The bias would be to a particular sector only. Any bias within a sector would be hidden until there were zillions of results in your data.

I disagree. Bias would not have to be sector dependent. If one slot were 'stickier' , or if the metal was askew to make it smaller, or bigger, any one number might have positive or negative bias. Actually, if one number slot was larger AT THE EXPENSE of its immediate neighbor, which is thus smaller, it would do exactly what you say it can't. Or, James Bond like, if one slot had a magnet and the ball had iron inside.....

Quote: FleaStiff

How a non-math person analyzes this:
Take the fewest appearing numbers and the most appearing numbers: Are they in the same or different sectors of the wheel?
It would be impossible to have a bias so fine that it favored one number but suppressed its neighbor. The bias would be to a particular sector only. Any bias within a sector would be hidden until there were zillions of results in your data.

I think a chi-squared test, which dwheatley explained how to do, is a good start. If the wheel passes that, which yours did, I would assume the wheel is okay. If the wheel fails a chi-squared test then graphing the results in order the numbers are laid out on the wheel would be appropriate, to test if the wheel is not balanced. Call it the eyeball test.

Quote: Wizard

I think a chi-squared test, which dwheatley explained how to do, is a good start. If the wheel passes that, which yours did, I would assume the wheel is okay. If the wheel fails a chi-squared test then graphing the results in order the numbers are laid out on the wheel would be appropriate, to test if the wheel is not balanced. Call it the eyeball test.

IF you are into Roulette spins and numbers...
Roulette Xtreme 2.0 is an excellent software to run Roulette sims and gather stats about spins.
It is free to use for 30 days then only costs $30.
A no brainer and no programming skills needed.
I have nothing to do with the company other than using their software.
http://www.uxsoftware.com/pages/index.html

Here is a snap shot of what the Wizard computed. The software does it all for you after you run the numbers from a text file into it. (Took 1 second to complete)Or you can enter them manually if you would like.



This is only 1 of many stat pages it contains.
Splits, corners, sectors, all adjustable and easy to copy and past to Excel or Word.
Good Luck

Good Luck

Quote: HKrandom

Try getting something that deviates more than 3 SD from the mean!

If you were to take 1363 spins from 1,000 different truly random wheels, about 70 of them would have a number at or above 3 standard deviations just by chance alone. This is because you are curve fitting for the standard deviation, and not determining it in advance.

Those 70 random wheels would have a chi-square, on average, of 45. This is very difficult to derive mathematically, and has to be determined through simulations.

Bias wheels are a bitch. 1363 spins is just not enough. The proper **minimum** methodogy is to take 1,500 spins, form a hypohthesis, then confirm in the next 1,500 spins. This will just indicate that the wheel is a good candidate. If the hypothesis confirms, you need another 3,000 spins to to re-confirm before you risk $1 of your money.

One way that helps to internalize this is to realize:

1) There must always be a best number. Just because you find a good number doesn't mean it is biased.
2) There must always be a best section. Just because good numbers are grouped in a section doesn't mean it is biased.
3) It is actually quite rare to have any data set evenly distruibed around the wheel. A Chi of 20 (flat) is just as rare as a Chi of 60.
4) There must always be 5 numbers that are the top 5 numbers. Just because you have found 5 numbers which - taken as a group - have a standard deviation above 4.0 doesn't mean they are biased. The chances you will find a wheel where the top 5 numbers will - as a group - be 4.0 standard deviations from the mean is 40%. 40 out of 100 random wheels will yield 5 numbers that as a group are at or above 4.0 standard deviations.

There always must be a set of top X numbers. Just because you have identified them doesn't mean anything.

Thank you
You have identified how I looked at the numbers. The analysis showed the required criteria for a biased wheel were not met, but the numbers fell in those two sectors of this wheel. So I was not sure if it was a biased wheel or not.

I will be in Macau next month so I will check out this same wheel again to see if there is anything similar still occurring.

Thanks to all posters for your excellent feedback

Quote: Pando

Thank you
You have identified how I looked at the numbers. The analysis showed the required criteria for a biased wheel were not met, but the numbers fell in those two sectors of this wheel. So I was not sure if it was a biased wheel or not.

I will be in Macau next month so I will check out this same wheel again to see if there is anything similar still occurring.

Thanks to all posters for your excellent feedback

It would be interesting to see all your recorded spins.
Then I could run the sim and see if those numbers or sectors actually stayed out front from the start.
I doubt that they all did.

Good luck next month.

I will record the numbers and make them available when I get back.

I always try to look for clusters, whether they are in dozens, or on sectors of the wheel. Sometimes it works, sometimes not.
Its a difficult task to do much without a computer but if I can see its worthwhile, I can take a laptop, and analyse the data on a daily basis, or even more frequently if necessary.

The hot numbers could then be found quite quickly, but the question is how relevant the recent history may be. Some argue it is of no use at all, but if there is a bias, or repeat clusters of numbers, I will find them.

Hi Guys
I am back from Macau and will write a trip report very soon.

On the first day I used the analysis of my 1363 spins recorded on my previous visit as a basis for betting this time around.
As you has all predicted, it was not viable. It actually worked for a time, about an hour at the specified table, but after that
the scenario changed and the numbers that had been working, suddenly stopped appearing. The best number last time round was 17 which started very promisingly, but then stopped completely. Some of the others never appeared at all.

I played for 2 hours and exhausted my bank roll of HK$4000. Then I started playing how I normally do and got back HK$3400 of it before quitting for the night.

I appreciate the feedback from all of you, but it was an interesting exercise to go through.

I have developed a formula for calculating the probability that some number or other will come up as many times as n in b spins on an unbiased roulette wheel. It currently is being discussed in the thread "New Probability Tables for Roulette IV." My formula is controversial. Some question its accuracy, some question that there are any biased wheels actually in service, and some question my ethics in offering tables for sale on Amazon for $25. The thread has been deleted twice and now is in the Free Speech Zone at the suggestion of one of the posters. I am probably the most hated person in the forums. Of course the tables are available free of charge to any member of the Wizard's forums who doesn't hurl insults at me.

That being said, I calculate the probability of a number coming up 51 times in 1,363 spins to be 0.11166 and conclude that this is not abnormal for a fair wheel. At 57 hits I would say that the wheel is biased at the 1% level of significance. I do not consider your sample too small.

> My formula is controversial.
I don't see why. Two plus two really is four and there should not be any controversy about this. Jokes perhaps (you know... how much do you want it to be) but not serious controversy.

> Some question its accuracy,

>some question that there are any biased wheels actually in service,
Maybe in Podunk, Iowa where they have not heard of computers yet and still use quill pens and bottles of ink for record keeping. However, in other casinos those darned bean counters keep track of those things.

> and some question my ethics in offering tables for sale on Amazon for $25.
Three legged tables?
> The thread has been deleted twice and now is in the Free Speech Zone at the suggestion of one of the posters.
You probably should have started it out there since there is no zone for Snake Oil.
>I am probably the most hated person in the forums.
No, but I think you would enjoy it if you were.
> who doesn't hurl insults at me.
Insults at you or at your insistence that two plus two ain't really four?
You got the numbers 1 thru 36, then you got either one green slot or you got two green slots. Assuming two green slots, that totals 38 and the ball is going to wind up in one and only one of those slots on the first spin. Second spin too. Even the third spin. Same thing. All night long, that croupier keeps telling the ball to wind up going somewhere else but it always falls into one of those slots. The ball doesn't listen to the croupiers instructions or to the hopes and fears of the players.
The ball doesn't have favorite slots or disfavored slots. Its just one out of thirty-eight. Thats all it is.

> I do not consider your sample too small.
Why is it that other number crunchers say that the sample is too small to make any valid statement about possible bias to the wheel from those results. Is it that all those other number crunchers go hatless but you wear aluminum foil?

Quote:

Fleastiff: Why is it that other number crunchers say that the sample is too small to make any valid statement about possible bias to the wheel from those results. Is it that all those other number crunchers go hatless but you wear aluminum foil?

You'll have to ask them! It is a general rule in law that someone who asserts a fact has the burden of proving it.

One of the objections to the tables is that they are for the range of 30 to 200 spins, which I considered to be a reasonable day's work for a player. There may not be any wheels in casinos with bias that be detected in as few as 30 spins but such home wheels do exist. I used to own one. A friend cleaned me out on it in about that many spins just by playing the most frequently occurring numbers.

Your comment about "snake oil" classifies you as a hostile correspondent. If you want to change that status you will have to choose your works more carefully.

Quote: scotty81

If you were to take 1363 spins from 1,000 different truly random wheels, about 70 of them would have a number at or above 3 standard deviations just by chance alone. This is because you are curve fitting for the standard deviation, and not determining it in advance.

Those 70 random wheels would have a chi-square, on average, of 45. This is very difficult to derive mathematically, and has to be determined through simulations.

Bias wheels are a bitch. 1363 spins is just not enough. The proper **minimum** methodogy is to take 1,500 spins, form a hypohthesis, then confirm in the next 1,500 spins. This will just indicate that the wheel is a good candidate. If the hypothesis confirms, you need another 3,000 spins to to re-confirm before you risk $1 of your money.

One way that helps to internalize this is to realize:

1) There must always be a best number. Just because you find a good number doesn't mean it is biased.
2) There must always be a best section. Just because good numbers are grouped in a section doesn't mean it is biased.
3) It is actually quite rare to have any data set evenly distruibed around the wheel. A Chi of 20 (flat) is just as rare as a Chi of 60.
4) There must always be 5 numbers that are the top 5 numbers. Just because you have found 5 numbers which - taken as a group - have a standard deviation above 4.0 doesn't mean they are biased. The chances you will find a wheel where the top 5 numbers will - as a group - be 4.0 standard deviations from the mean is 40%. 40 out of 100 random wheels will yield 5 numbers that as a group are at or above 4.0 standard deviations.

There always must be a set of top X numbers. Just because you have identified them doesn't mean anything.

100 of 1000 will have 1 number with 3 st dev, it is no 70/1000, tell me if I´m wrong. 99,73% is to be inside 3sd, 0,027 is the remains for 3 or more sd, 27/10000.

To find 3 sd in any location of the wheel is 27x37=1000/10000 or 1/10, this goes to our next sentence.

How do you compare "curve fitting" with real facts?

"4) There must always be 5 numbers that are the top 5 numbers. Just because you have found 5 numbers which - taken as a group - have a standard deviation above 4.0 doesn't mean they are biased" Here is there difference between curve fitting and real facts. The same event might mean different degrees of certanty for viewers.
This sample could be my 3rd analized sample but the 1er for other player, for me 4sd is in this case 99,95% of confidence, for other is 96%.

"The chances you will find a wheel where the top 5 numbers will - as a group - be 4.0 standard deviations from the mean is 40%. 40 out of 100 random wheels will yield 5 numbers that as a group are at or above 4.0 standard deviations." How did you calculate it?

Quote: ybot

100 of 1000 will have 1 number with 3 st dev, it is no 70/1000, tell me if I´m wrong. 99,73% is to be inside 3sd, 0,027 is the remains for 3 or more sd, 27/1000.

I haven't read the thread but am chiming in to point this out:
99.73% = 0.9973
1 - 0.9973 = 0.0027 not 0.027
0.0027 = 27/10000 not 27/1000

Sorry, you´re right, it is 27/10000.

I tried to place the correct % because it is nor 99,7 but 99,73(not 3/1000, yes 2,7/1000).

Quote: statman

I calculate the probability of a number coming up 51 times in 1,363 spins to be 0.11166 and conclude that this is not abnormal for a fair wheel. At 57 hits I would say that the wheel is biased at the 1% level of significance. I do not consider your sample too small.

Statman arrives at the same conclusion as the Wizard but requires far less data. What is his formula and why is he so unpopular? I think he may be on to something.