About probability of playing roulette

[KFC]

Hi there,
I am reading a book about statistics and probability. In one chapter, the author use roulette as an example to explain the use of central limit theorem. He says for ordinary roulette, there are 38 numbers (including one 0 and 00). The chance of occurrance of 17 is 1/38. But for a wrapped wheel (assume extreme warp in section 17), the change of getting 17 will be increased. So statistically, how to tell if a wheel is wrapped? The following is excerpted from the book

In order to such a decision you must account for chance variation. One way to do this is by computing the standard deviation and using the central limit theorem.It turns out that even for this extremely warped wheel you need to observe about 2000 spins to get enough separation to effectively rule out change. In 2000 spins, the three-SD (three times of standard deviation) range around the average number of 17's for the warped wheel (average=105.26; 3SD=29.96) does not overlap the three-SD range for an ordinary wheel (average=52.63; 3SD=21.48). In other words, the number of occurrences of 17 with this extremely warped wheel will most likely be outside the three SD range of an ordinary wheel, so you could confidently rule out chance variance.

Sorry for the long information. Here are what I confuse
1) Why we have to rule out the "chance variation"? My opinion is to make sure the biased outcome of roulette is because of the wrap not because the chance, is that right?

2) I don't understand why the author say because of no overlap of two normal distribution, we could confidently rule out chance variance? How to understand that? What happen if two curves do overlap?

 

[Office_Shredder]

I'm not really sure what you mean by a wrapped wheel, but let's take a look at the rest of the post:

Imagine I have two coins. I flip one and it comes up heads. I flip another one and it comes up tails. "Look!" I cry. "I have two different coins; one gives heads when flipped, the other gives tails."

You probably agree that would be a silly leap of faith from a single coin flip. We have to account for chance variation because things that take values randomly are supposed to be different, even if they have the same distribution. So you can't just say that two sequences of events are different so must be controlled by different distributions - instead you have to show in some way that they are so different that they cannot be controlled by the same distribution.

In this case we consider one sequence which is spinning a normal wheel, and another one spinning a wrapped wheel. By measuring how often 17 comes up, it turns out that the number of occurrences is different enough between the two wheels that we can confidently say which wheel it is based on how often 17 comes up (remember, we could be wrong! The sequence of spins that we get could look like it comes from the wrapped wheel, but simply be a low probability sequence from the unwrapped wheel)

The normal curves are always going to overlap because they're non-zero everywhere, but the part within 3 standard deviations of the mean don't overlap. Since it is unlikely for the number of 17's to lie more than three standard deviations from the average number of 17's this means that with high probability we can pick the right type of wheel based on the number of 17's.

If the two parts of the curve did overlap, we would have a much harder time deciding which wheel it was, especially if the number of 17's landed somewhere in the overlap

 

[Mark44]

Try to be consistent in your description of the roulette wheel. You have described it as "wrapped" in four places and "warped" in four places.

I think you mean "warped" so that the wheel is not consistently flat. "Wrapped" and "warped" are entirely different words.