Walkinshaw30t
Joined: Apr 11, 2013
• Posts: 91
April 11th, 2013 at 6:35:08 AM permalink
I can not remember the math to work out the answer and if someone could explain the math it would be greatly appreciated!!
Time will tell
dwheatley
Joined: Nov 16, 2009
• Posts: 1246
April 11th, 2013 at 6:42:13 AM permalink
=BINOMDIST(350,850,18/38,TRUE)

will get you the probability of 350 blacks or less in 850 spins, assuming 00 roulette. 1 in 6108.
Wisdom is the quality that keeps you out of situations where you would otherwise need it
Doc
Joined: Feb 27, 2010
• Posts: 6935
April 11th, 2013 at 6:55:39 AM permalink
(1) Assuming this is about roulette, how many green numbers are there on this particular wheel?
(2) Do you mean exactly 350 blacks, at least 350 blacks, or something different?
(3) Is there any reason to suspect anything other than a completely unbiased wheel?
(4) Is this a homework problem?
Walkinshaw30t
Joined: Apr 11, 2013
• Posts: 91
April 11th, 2013 at 7:04:33 AM permalink
Cheers yes sorry I should have been more clear its regarding roulette.
Single zero roulette wheel
350 or less in 850 spins
It is more so for me to learn the math- could you tell how many standard deviations away from expected this is?
Is it possible for this to happen on a fair wheel?
Time will tell
Doc
Joined: Feb 27, 2010
• Posts: 6935
April 11th, 2013 at 9:10:07 AM permalink
Two important issues come up in my attempt to answer your question. One is that I am so rusty on my statistics that I'll likely make an error and someone else will have to correct me. The other is that it is cumbersome on this site to write mathematical expressions in a post so as to assist in explaining the stuff.

Before I begin, I will just answer your last question. Yes, it is possible for it to happen on a fair wheel. For that matter, it is possible for all 850 of the spins to come up black on a fair wheel. Statistics just lets you consider the likelihood of outcomes and develop some confidence as to whether the wheel is really fair or not.

Here's my attempt at the math:

850 spins → n = 850
single-zero roulette → p = 18/37
Probability of 350 or fewer black:

That expression is quite difficult to evaluate on most hand calculators. That's why dwheatley was suggesting using a function on an Excel spreadsheet. Since you have now clarified that this is single-zero roulette, I think the answer is P=7.209x10^-6, or about 1 chance in 138,711.
CrystalMath

Joined: May 10, 2011
• Posts: 1791
April 11th, 2013 at 9:24:16 AM permalink
Quote: Doc

Before I begin, I will just answer your last question. Yes, it is possible for it to happen on a fair wheel. For that matter, it is possible for all 850 of the spins to come up black on a fair wheel. Statistics just lets you consider the likelihood of outcomes and develop some confidence as to whether the wheel is really fair or not.
...
1 chance in 138,711.

Of course, they could all come up black. Although the question was "is it possible," the intent of the question is "is there statistical evidence that the wheel is unfair?"

Personally, I would not consider 1:139k to be evidence of an unfair wheel. Top award on most slot machines is 1:17 million.
I heart Crystal Math.
Doc
Joined: Feb 27, 2010
• Posts: 6935
April 11th, 2013 at 9:29:48 AM permalink
I forgot to answer your other questions. I think this covers them:

EV = np = 850 * (18/37) = 413.51

Var = np(1-p) = 850 * (18/37) * (19/37) = 212.34

St. Dev. = sqrt(Var) = sqrt(212.34) = 14.57

So 350 is about 4.36 standard deviations below the expected value.

(O.K., now, somebody check my math!)

Whether this is evidence of an unfair wheel depends quite a bit upon whether this was a random group of 850 spins or a group that was specifically selected out of millions of spins, with this selected group having few black results and other groups having many more. I used to like pointing out situations where people thought they were dealing with a random selection when they were not.
Walkinshaw30t
Joined: Apr 11, 2013