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science forum beginner

Joined: 23 Jun 2006
Posts: 1

Posted: Fri Jun 23, 2006 2:27 pm    Post subject: [Help] Information needed

Hello all,

I have two simple basic questions:

1- How do I determine if a coin is fair or not. Assuming for example, I
get 5500 heads out of 10000 coin tosses.
2- How do I determine the probability distribution of given data
points?

I need is a general guideline as how to approach these types of
problems. BTW these are not homework problems I am a practicing
engineer with a very basic knowledge of statistical theory. Any help or
references will be appreciated.

Frank
Bill H
science forum beginner

Joined: 11 May 2006
Posts: 9

Posted: Fri Jun 23, 2006 3:43 pm    Post subject: Re: Information needed

 Quote: Hello all, I have two simple basic questions: 1- How do I determine if a coin is fair or not. Assuming for example, I get 5500 heads out of 10000 coin tosses. 2- How do I determine the probability distribution of given data points? I need is a general guideline as how to approach these types of problems. BTW these are not homework problems I am a practicing engineer with a very basic knowledge of statistical theory. Any help or references will be appreciated. Frank

OK, I know this one! With that many N you can use the normal
approximation to the binomial. Normalize your sample proportion by
subtracting the mean = X/N = 5500/10000 and dividing by SD under the
null value of p = .5 for a fair coin = sqrt(.5(1-.5)/N) = .005 so that
Pr(p not equal p0) = 2*Pr(p>p0) [by symmetry of normal]. Then
normalizing, 2*Pr( (p -.5)/.005 > (.55 - .5)/.005) = 2*Pr(Z > .05/.005)
= 2*Pr(Z > 10) = very very small number so that one may conclude on the
basis of that sample that it is very unlikely the coin is not fair
(where Z is a standard normal variate).
Bill H
science forum beginner

Joined: 11 May 2006
Posts: 9

Posted: Fri Jun 23, 2006 5:00 pm    Post subject: Re: Information needed

Bill H wrote:
 Quote: fadamster@gmail.com wrote: Hello all, I have two simple basic questions: 1- How do I determine if a coin is fair or not. Assuming for example, I get 5500 heads out of 10000 coin tosses. 2- How do I determine the probability distribution of given data points? I need is a general guideline as how to approach these types of problems. BTW these are not homework problems I am a practicing engineer with a very basic knowledge of statistical theory. Any help or references will be appreciated. Frank OK, I know this one! With that many N you can use the normal approximation to the binomial. Normalize your sample proportion by subtracting the mean = X/N = 5500/10000 and dividing by SD under the null value of p = .5 for a fair coin = sqrt(.5(1-.5)/N) = .005 so that Pr(p not equal p0) = 2*Pr(p>p0) [by symmetry of normal]. Then normalizing, 2*Pr( (p -.5)/.005 > (.55 - .5)/.005) = 2*Pr(Z > .05/.005) = 2*Pr(Z > 10) = very very small number so that one may conclude on the basis of that sample that it is very unlikely the coin is not fair (where Z is a standard normal variate).

Eh, change that to "very unlikely the coin IS fair", ie. likely the
coin is NOT fair, ie. away from the null.
Steve Nelson
science forum beginner

Joined: 28 May 2006
Posts: 2

Posted: Fri Jun 23, 2006 5:29 pm    Post subject: Re: Information needed

 Quote: I have two simple basic questions: 1- How do I determine if a coin is fair or not. Assuming for example, I get 5500 heads out of 10000 coin tosses. 2- How do I determine the probability distribution of given data points? I need is a general guideline as how to approach these types of problems. BTW these are not homework problems I am a practicing engineer with a very basic knowledge of statistical theory. Any help or references will be appreciated.

Wikipdeia is a big help on this. One page for the binomial
distribution and another for the normal curve.

http://en.wikipedia.org/wiki/Binomial_distribution
http://en.wikipedia.org/wiki/Normal_curve

The binomial page says that for n = 10000 and p = .5, the variance is
2500. The standard deviation is the square root of that, or 50.The
normal curve page says that 68% of the flips should be in the band
between 5000 - 50 and 5000 + 50, or 4950 to 5050.
The numbers you posted are about ten standrad deviations out so it is
almost certainly a biased coin.
illywhacker
science forum beginner

Joined: 25 Jul 2005
Posts: 9

Posted: Wed Jul 05, 2006 11:43 am    Post subject: Re: Information needed

 Quote: 1- How do I determine if a coin is fair or not. Assuming for example, I get 5500 heads out of 10000 coin tosses. 2- How do I determine the probability distribution of given data points? I need is a general guideline as how to approach these types of problems.

I suggest you go and read E. T Jaynes' book, 'Probability theory: the
logic of science'. This will not only tell you how to solve this
probabilistic problems in general, and provide you with a unified way
to approach inference problems.

You might want to consider what you mean by saying 'a coin is fair',
question you are asking, and hence cannot expect a clear answer (of
course, there are always textbook 'solutions' to be trotted out).

illywhacker;
Philipp Pagel
science forum beginner

Joined: 05 Jul 2006
Posts: 1

Posted: Wed Jul 05, 2006 12:13 pm    Post subject: Re: [Help] Information needed

 Quote: 1- How do I determine if a coin is fair or not. Assuming for example, I get 5500 heads out of 10000 coin tosses.

Others have made helpful replies to you actual questions. I'd just like
to add a link to a paper on the topic of biasing coins that i found quite
amusing:

http://www.stat.berkeley.edu/~nolan/Papers/dice.pdf

cu
Philipp

--
Dr. Philipp Pagel Tel. +49-8161-71 2131
Dept. of Genome Oriented Bioinformatics Fax. +49-8161-71 2186
Technical University of Munich
http://mips.gsf.de/staff/pagel
John Bailey

Joined: 05 May 2005
Posts: 72

Posted: Thu Jul 06, 2006 11:39 am    Post subject: Re: [Help] Information needed

On Wed, 5 Jul 2006 12:13:17 +0000 (UTC), Philipp Pagel
<pDOTpagel@gsf.de> wrote:

 Quote: Others have made helpful replies to you actual questions. I'd just like to add a link to a paper on the topic of biasing coins that i found quite amusing: http://www.stat.berkeley.edu/~nolan/Papers/dice.pdf

Thanks, I needed that.

The limiting case of a biased coin is the fat coin--where the
thickness is large enough for landing on its edge to be a significant
possibility. (Think cat food can)
http://tinyurl.com/m7ta5 is a link to theoretical pronouncements which
(quoting)
There are two conditions that have to be met for a coin to yield 3
equal probablities for landing on one of three surfaces: the head
face, the other face or the edge.

1) The solid angle subtended by each surface needs to be equal. Said
another way, if you think of a sphere enclosing the coin, the areas on
the sphere that are associated with each surface must be equal. This
is straightforward solid geometry and the equations for calculating
this condition are given in the quote.

2) The potential energy, ie, the weight times distance of the center
of gravity of the coin from the landing surface must be equal for all
three orientations.
(end quote)

For a more normally shaped (thinner) coin which could be biased, the
two conditions for fairness should be approximately met for the two
coin faces.

extremely wide range of variation away from these two nominal
conditions within which a coin's bias remains statistically
insignificant.

The other factor in a biased coin that was cited is the flipping
technique. An unbiased coin is less sensitive to flipping variation
experimental support to that conclusion. See http://tinyurl.com/pfxyx

de Moivre is rolling in his grave!
infarom@hotmail.com
science forum beginner

Joined: 01 Jun 2006
Posts: 2

Posted: Mon Jul 10, 2006 1:26 pm    Post subject: Re: Information needed

 Quote: Hello all, I have two simple basic questions: 1- How do I determine if a coin is fair or not. Assuming for example, I get 5500 heads out of 10000 coin tosses. 2- How do I determine the probability distribution of given data points? I need is a general guideline as how to approach these types of problems. BTW these are not homework problems I am a practicing engineer with a very basic knowledge of statistical theory. Any help or references will be appreciated. Frank

A very good book on probability basics, calculus and applications for
non-mathematicians has just been released: "Understanding and
Calculating the Odds: Probability Theory Basics and Calculus Guide for
Beginners, with Applications in Games of Chance and Everyday Life". You
can find info and a free e-sample at http://probability.infarom.ro .

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