Quote:MathExtremistWhat if I said that there's a roulette wheel in Nevada right now where the number 00 came up 9 times in the past 200 spins. Yes or no: is the wheel biased?

Maybe. But not likely.

Quote:MathExtremistI want to know what his tables say about the scenario where 00 comes up 9 times in 200 spins. Is that wheel biased or not?

From statmans tables it shows a 0.81660 probability.

So I do not think that would show a bias.

My sim shows 0.8358

But you already showed that his formula is about a particular number and not just any number.

ME, just give statman a quick PM and I am sure he would email to you his Excel worksheet and Kindle file of tables as he did for me.

I am in the middle of a large math and music project and just do not have the time, at this time, to pursue this matter.

I am always interested in unique math solutions to interesting math problems, that is the only reason I became involved.

Enjoy

Quote:guido111From statmans tables it shows a 0.81660 probability.

So I do not think that would show a bias.

My sim shows 0.8358

But you already showed that his formula is about a particular number and not just any number.

I'm interested in what is statman's interpretation of that number (whether it's 0.81 or 0.83) and what it means to his understanding of whether the wheel is biased.

Quote:guido111I am in the middle of a large math and music project and just do not have the time, at this time, to pursue this matter.

To completely hijack the thread, that sounds really cool. Are you doing something with algorithmic composition?

Quote:MathExtremistI'm interested in what is statman's interpretation of that number (whether it's 0.81 or 0.83) and what it means to his understanding of whether the wheel is biased.

To completely hijack the thread, that sounds really cool. Are you doing something with algorithmic composition?

Statman uses a 150 spin and 9 times hit as an example in his document that does not show a biased wheel but he really does not get into what would be a wheel bias as far as I understand.

He continues with "We would have to see 11 or 12 on the same number to call it unusual."

0.36028 = 9 hits

0.05660 = 11 hits

0.01806 = 12 hits

FYI: I do musical score arrangements and have done some interesting algorithmic compositions in the past but not currently.

It is fun stuff with the quality of today's digital keyboards.

Quote:guido111Statman uses a 150 spin and 9 times hit as an example in his document that does not show a biased wheel but he really does not get into what would be a wheel bias as far as I understand.

He continues with "We would have to see 11 or 12 on the same number to call it unusual."

0.36028 = 9 hits

0.05660 = 11 hits

0.01806 = 12 hits

So let's make it unusual: suppose you observe a roulette wheel for 150 spins and see the 00 show up 15 times. Now what? Does he conclude that the wheel is biased? Does he conclude that the likelihood of 00 appearing on the next spin is greater than 1/38?

If there is a wheel bias, for example because the slots are not all the same size, more likely than not, the error is caused by a misplaced single divider. As such, on that wheel, there would be a bias for a number, as well as a bias against the neighbor.

Additionally, if there is a wheel bias because its out of balance, then it will be biased for range(s) of numbers and/or biased against range(s) of numbers.

I doubt that his formulas account for either of these types of multiple biases.

Dealer ID | Wheel direction | Wheel RPM | p(theta between 0 and 45 degrees) | p(theta between 45 and 90 degrees) | ...

where theta is the angle between the spot on the wheel when the ball was released and where it lands. On a fair wheel, fairly dealt by a croupier, each probability will be 1/8 (give or take for the rounding issues). If you find one that's significantly greater than 1/8, there's a reasonable chance of a bias, and there's also a credible physical cause. It's not one that you'd ever detect by counting outcomes because theta isn't computed based on the actual outcome -- just the distance between the outcome and the release point.

Granted, you still need to be able to make bets after the ball drop, but you still can in most casinos. So, for example, if you have a dealer who tends to hit the octant between 135 and 180 the most when the wheel is spinning at a given speed, and you see the wheel spinning at that speed, wait for the ball to release and then put your money on those 5 numbers. Consider this a relative, table-lookup version of Eudaemonic Pie.

Some of it eventually may reappear on the web site of the

Rancocas Valley Journal of Applied Mathematics

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Many thanks to those who have been helpful.