mustangsally
Joined: Mar 29, 2011
• Posts: 2463
October 6th, 2011 at 3:50:21 PM permalink
Quote: statman

Who is Right? (Or who is closer to the correct value?)

This question is to be found on http://wizardofodds.com/askthewizard/roulette.html

What are the chances of a dealer hitting 5 of the same number in 10 spins of the roulette wheel? -- Spence from Red Deer, Alberta

Wiz: The chances of any number occurring exactly 5 times in 10 spins in a double-zero roulette game could be closely approximated by 38*combin(10,5)*(1/38)5*(37/38)5 = 1 in 359,275.

Apparently he has a way of figuring these things out, but we are far apart. His answer is 0.000002783, My answer is 0.00011.
We also are apart by a factor of 100 on a subsequent question: 6 hits in 1000 spins. My tables say the probability is 0.00005

I do not know who is closer to the exact answer. Where is the exact answer?

The Wizards formula and his 1 in do not match. No one is perfect all the time. He also states it is approximated.

38*combin(10,5)*(1/38)^5*(37/38)^5 = 0.000105769 or 1 in 9454.6 (Maybe he was changing a diaper at the time and was distracted by a phone call)
Your table answer is 1 in 9090.9
My 10 million sim is 0.0001087 or 1 in 9199.6

I say we all win a prize!

There is no doubt that the exact math formulas for these type of questions is very challenging.
Since you use Excel and know how to solve this with inclusion-exclusion, a process very time consuming and like you said before, possibility of errors,
why not just set up a macro in Excel to run the calculations? That way you can correct any minor mistakes you may find and you will finally have the exact answers you are looking for instead of "just close" answers that may not always be that close.

You will be the first.
I say go for it!

added a few minutes later:
From Statman's previous previous post:
"Would the above be worth the extra time and trouble to program
and execute seeing that the difference between the approximate and exact values appear to be
only about 1% "
I say yes! Unless you know that ALL your results are that close, better to be perfect if given the choice.
I work with DNA researchers, no I am not one of them, and they freak out when something in their data is off by .001%. Of course they deal with sample sizes of a few billion so I do understand them wanting to be exact in their calculations.
I Heart Vi Hart
mustangsally
Joined: Mar 29, 2011
• Posts: 2463
October 6th, 2011 at 4:19:20 PM permalink
Quote: statman

Who is Right? (Or who is closer to the correct value?)

This question is to be found on http://wizardofodds.com/askthewizard/roulette.html

We also are apart by a factor of 100 on a subsequent question: 6 hits in 1000 spins. My tables say the probability is 0.00005

I missed the last question from the WoO site.
"I played the same number 1000 spins in a row on a 0,00 wheel and hit 6 times. What are the chances of hitting 6 or fewer times in this scenario? Thanks, Bill K."
This is just a binomial distribution question. And Bill is looking for a "specific" number not any number. Right?
In Excel =BINOMDIST(6,1000,1/38,TRUE)
1.77565E-06 or 1 in 563175.4617
The Wizard continues "So the answer is 0.00000177564555, or 1 in 563175. I hope this didn't happen at an Internet casino"
I Heart Vi Hart
boymimbo
Joined: Nov 12, 2009
• Posts: 5993
October 6th, 2011 at 5:05:58 PM permalink
The point of the binomial and multinomial distributions are that they are inclusive. The formulas and theories take into account that some other outcome will occur. Therefore they are valid. The odds of ANY number hitting 5 times in 10 spins is 1/38 x the odds of 1 number hitting 5 times in 10 spins.

So, when you ask, what are the odds of number x hitting n times in b spins, it's 1/38 of the odds of any number hitting n times in b spins.
----- You want the truth! You can't handle the truth!
statman
Joined: Sep 25, 2011
• Posts: 95
October 7th, 2011 at 6:47:49 AM permalink
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CrystalMath

Joined: May 10, 2011
• Posts: 1806
October 7th, 2011 at 8:32:03 AM permalink
Quote: statman

I hesitate to find fault with the Wizard's work. It would be like criticizing God.

If you can find a problem with his work, he is very humble and appreciative. If he wanted to back you up, he would be on here.

As for you insulting people (such as MathExtremist), that is very distasteful.
I heart Crystal Math.
boymimbo
Joined: Nov 12, 2009
• Posts: 5993
October 7th, 2011 at 12:45:54 PM permalink
The wizard's answer relies on the binomial distribution, which is a foundation to statistics a long time ago. He states it's a good approximation because actually, the multinomial distribution applies.

Suppose a multinomial experiment consists of n trials, and each trial can result in any of k possible outcomes: E1, E2, . . . , Ek. Suppose, further, that each possible outcome can occur with probabilities p1, p2, . . . , pk. Then, the probability (P) that E1 occurs n1 times, E2 occurs n2 times, . . . , and Ek occurs nk times is
P = [ n! / ( n1! * n2! * ... nk! ) ] * ( p1n1 * p2n2 * . . . * pknk )

where n = n1 + n2 + . . . + nk.

You ask, what is the probability that a result is on any number 5 times in 10 spins, and the Wiz used the binomial distribution to calculate the result for the number 1 and then multiplied it by 38. It's an approximation because it is possible for say example 00 to occur 5 times and 0 to occur 5 times. He multiplies the result by 38 to get the approximation.

The correct analysis would use the mathemically sound multinomial distribution, but the number of iterations required on a roulette wheel to get all possible answers would be a very long one indeed, not impossible, but definitely programmable, and long.

The point that mathextremist, crystalmath, and I are making is that there is nothing new here.
----- You want the truth! You can't handle the truth!
statman
Joined: Sep 25, 2011
• Posts: 95
October 7th, 2011 at 4:22:15 PM permalink
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statman
Joined: Sep 25, 2011
• Posts: 95
October 7th, 2011 at 4:33:34 PM permalink
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7craps
Joined: Jan 23, 2010
• Posts: 1977
October 7th, 2011 at 5:00:11 PM permalink
Quote: statman

For those who have access to Jstor, the link is http://www.jstor.org/pss/2287733
If I can get the other pages, I'll post them as well. If anyone is interested in Ethier's book, let me know. Discount what Scarne says. He didn't believe in Thorp's card counting strategy.

S.N. Ethier's Book
The Doctrine of Chances 2010
chapter 13.2
"Suppose we have “clocked” a roulette wheel for n coups, and the most
frequent number has occurred a proportion 1/25 of the time. Is it a biased
wheel?"

Excellent reading and he even appears to answer the question

He is a top notch Math Professor

added: in the same book
"13.16.
Number of coups needed to conclude that a favorable number exists.
We saw that, if the most frequent number in n coups has occurred
a proportion 1/25 of the time, we can conclude with 95% confidence that
one or more favorable numbers exist if n ≥ 1,618."
winsome johnny (not Win some johnny)
FleaStiff
Joined: Oct 19, 2009
• Posts: 14319
October 7th, 2011 at 5:03:13 PM permalink
Quote: statman

00000

Okay. Much of this thread consists of five zero posts which appears to reflect some Wiki Memory Wipe of someone who wanted to take his toys and go home.

The rest of the posts seem to be something about multinomial distributions are arguments of numbers hitting and numbers not hitting and what can considered proof of something versus evidence consistent with it, but not proving it.

I'm lost on much of the math anyway.

May I propose a practical trial?

Will someone who knows what he is talking about and does not believe in the utility of these much touted tables simply get a set of the tables and sit down at one of those cheapie roulette wheels that require a grand total of fifty cents on inside bets or something. Clock the wheel for an hour, then play according to the tables for an hour... and let me know whether you win or lose.