# I Are these alleged equal probabilities really equal?

1. Jun 24, 2017

### borson

Let me explain myself.
In the roulette game, you can bet on dozens, that means, betting to 12 numbers which are placed separately along the wheel. (12 out of 37 total numbers).
But I had this curious doubt. if you bet on a sector of the roulette, composed by 12 numbers.... the likelihood is really the same as before?
I know that I'd still be 12/37, but from my point of view, it is easier to obtain that sector as the outcome as it is, physically, a part/section of the roulette.
Let's say we want to do a Martingale (increasing the wager each time that we fail)
Wouldn't it be easier for a dozen than for a sector to be more many times without turning out?
Because on a dozen, if it will turn out or not, basically depends on randomness; amount of rebound and so on.
But on a sector, it depends in more general physical factors.

2. Jun 24, 2017

### Orodruin

Staff Emeritus
This is the kind of thinking that makes you go broke. If you spin such that the outcome is evenly distributed there is no difference. It may be easier to cheat if you are the one spinning the wheel and control the rotation and ball speed.

3. Jun 24, 2017

### UsableThought

What scenario you imagining here? Remember, per spin of the wheel, the ball can only drop once; and only into a pocket, not into a "sector"; so the sector gains no advantage from being "bigger" than a pocket or from having "more than one pocket together." And if the wheel is "fair," that is, does not favor any particular pocket, it will also be fair in not favoring any particular sector of two or more contiguous pockets.

This concept of "fairness" is actually more interesting than it might seem at first, as it is not only a requirement for all the classical gambling games - dice, cards, roulette - but was later incorporated into the mathematics of classical probability, such that all single outcomes are presumed equally likely - not just in gambling games but any phenomenon under consideration. This allows uncertainty for sufficiently complex events to be described without regard to causality - e.g. with roulette we don't need to worry about precise Newtonian descriptions of exactly how fast the wheel turned each time, or exactly how the ball bounced along the track before finding a pocket, etc. And from what I have been reading, this deliberately narrowed view of uncertainty not only became extremely useful to science, but facilitated the emergence of mathematical statistics by around 1930 or so. My source for this is a very interesting book, Willful Ignorance: The Mismeasurement of Uncertainty, by Herbert Weisberg.

Anyway if pondering roulette wheels etc. interests you & you want to learn more, just read up on classical probability. A nice little teach-yourself book is the one put out by Dover: Introduction to Probability, by John E. Freund. I read through it & did the exercises maybe 15 or 16 years ago & found it great fun. Of course there are lot of other good books out there also.

(FYI, a deck of cards is made "fair" by requiring all cards to be equally unrecognizable from the back, the deck to be sufficiently shuffled such that distribution can be considered random, and no cards missing or repeated.)

Last edited: Jun 24, 2017
4. Jun 24, 2017

### borson

Yes I do know that probabilities should be the same (12/37 in both). What I mean is that the sector would be physically easier to turn out. Because
1st: A determined dozen, to be turned out, depends on more specific and caothic variables such as the size of the frets or the final rebounds. On the other hand a sector would depend on wider variables, such as the ball's and wheel's velocities.
2st: Although the amount of numbers we are betting on are the same, we are not taking into account the physical circumstances to make that statistic.

I do not know if I am explaining myself well. Let me put an example.
Imagine that you are playing dards. I think it would be easier for you to nail them on a determined sector of the dartboard than on specific numbers distributed all along the target. (occupying both, in total, the same amount of space on the dartboard)

5. Jun 24, 2017

### Orodruin

Staff Emeritus
If you aim for it yes, but that is not the point of a roulette wheel. The entire construction is intended to spread the outcomes as evenly as possible. Even if it wasn't, are you going to tell the croupier to aim for your section of the wheel?

6. Jun 24, 2017

### borson

Well I had in mind electromechanical roulettes, and not those worked by croupiers.
The construction is intended to be as much random as possible, and it suffices well its purpose in the long term.
However, sometimes we can see large series of for instance reds or blakcs, it is not so uncommon to see, let's say, 14 reds in a row.
With dozens that series would be even larger, we could perfectly see 20 times without one of them having been turned out.
What I am considering then is, if performing a martingale on a sector would be better or even really profitable than on a dozen.
As for the reasons stated before, it would be more difficult to have a so large bad streak betting on a sector.

7. Jun 24, 2017

### FactChecker

Any number is part of a sector, whether you picked that number as one of a random 12 or as part of that sector. So how would the ball know which way you picked that number? If you think that the number's probability depends on how it was picked, you are giving the ball credit for more intelligence than it deserves.

8. Jun 24, 2017

### borson

The ball's probability would depend on the rotor's and ball's velocities.
I am not talking about betting on only one number but on a sector of 12 numbers rather than on a dozen.

9. Jun 24, 2017

### borson

I am not saying that a sector is more likely than a dozen, but that it is less likely to have a bad streak betting on a sector than on a dozen

10. Jun 24, 2017

### FactChecker

The probability of any number can not depend on how it was selected by you. So none of the probabilities of any of the 12 numbers depend on whether they were selected as part of a section or not.

That being said, there may be numbers or sections that are more likely in a wheel that is not fair. In that case, you might change the probabilities by making different selections. But that is not the situation with a fair wheel.

11. Jun 24, 2017

### Orodruin

Staff Emeritus
Again, this is exactly the kind of thinking casinos profit from. If the wheel is fair, then the probabilities are equal and there is nothing that gives one or another combination of 12 numbers less probability of a cold-streak. Take any selection of 12 numbers and you will have the same distribution for the number of times you need to spin the wheel before winning. If the wheel is fair, then your "aiming at a sector on the dart board" analogy is clearly flawed.

12. Jun 24, 2017

### StoneTemplePython

One interesting historical note is that in practice, with the right technology you could (still can?) break the approximation of roulette wheels being fair.

Claude Shannon and Ed Thorpe did exactly this, culminating in the world's first wearable computer.

https://www.cs.virginia.edu/~evans/thorp.pdf

Casinos would not take kindly to this kind of thing and it is not legal now. That said, I am a bit surprised that there isn't an app for this for my iphone.

13. Jun 24, 2017

### UsableThought

I agree w/ @Orodruin on the kind of thinking your quotes here represent - it's the sort of fantasy view of probability that the gambling industry is happy for you to believe in. Maybe you have been reading too many advice articles on "how to beat roulette"? You're looking for what gamblers call a "system" - you and Dostoyevsky both!

In your first comment quoted above, you seem to be suggesting that bettors can avoid bad streaks by carefully manipulating sequence or type of bets, e.g. using sectors and so on, so as to beat the law of large numbers. This plus your second comment, about short-term runs of red & black somehow seeming significant, leads me to recommend you read Wikipedia's article on the 'gambler's fallacy'. Among other things, the article describes the psychology involved:
If you educate yourself from textbooks rather than gambling pages on the Internet, all these clever lovely fantasies will disappear; on the other hand, developing an understanding of probability has pleasures of its own.

Last edited: Jun 25, 2017
14. Jun 24, 2017

### UsableThought

Nice catch to remember this.

One thing that is interesting, reading the actual paper, is that Thorpe's starting idea was opposite what one might have expected: rather than identify & exploit unfair wheels, he began with the assumption (which later had to be modified) that the typical casino wheel is fair - thus making prediction possible. He puts it nicely at the start of the article:

I believed that roulette wheels were mechanically well made and well maintained. With that, the orbiting roulette ball suddenly seemed like a planet in its stately, precise and predictable path.

I set to work with the idea of measuring the position and velocity of the ball and rotor to predict their future paths and from this where the ball would stop. Such a system requires that bets be placed after the ball and rotor are set in motion.​

Kind of like the Johannes Kepler of roulette, eh?

Last edited: Jun 25, 2017
15. Jun 25, 2017

### borson

Yes, I knew about the gambler's falacy. I am stating this question because it impacted me that dozens had the same probability as sectors, due to the physical reasons already stated.
What you are saying about the biased roulettes is interesting. I have a doubt about that:

I have analysed some roulette spins with sony vegas in order to get the time for each turn of the ball.
It is supposed that if the time that takes for the ball to complete a turn is higher, it means that its velocity is faster and thus the time at which it will fall off from the rim will be lower too.

For instance, let's say that the ball completes a turn in 0,5 seconds. Thus, it is supposed to be faster than when the ball completes the turn in 0,6s. Hence, the former is supposed to last more time in falling into the wheel than the 2nd. For instance, the 1st should last.. let's say.. 22s. and the 2nd 20s. That would be the normal.

However in the spins that I have analyzed (from several roulettes) I always get weird results.
Some "turn-times" are lower than others, and their final times are also lower than those of the other lower turn times. I mean, it is like if the time turn of 0.6s lasted more time in falling than the turn-time of 0.5s. Which wouldn't make sense.

Due to this and other strange things (some "turn-times" which are quite different, have the same or similar final times) I do not see that there is any nonlinear relationship or that it follows any kind of function.

I have been reading on internet some papers and articles and, it seems that if there is any kind of inclination, no matter how small that angle is, there are zones of the roulette in which the ball will fall and it would lead to biases. I do not know if it could be related to the outcomes that I have obtained.

What do you think?
Thank you all for your help!

16. Jun 25, 2017

### UsableThought

Did you read the Thorpe paper that was linked to? They looked at the physical track and the behavior of ball and wheel very methodically. So that might be your best source of information at the moment.