Pi and Probability

Date: 12/21/97 at 19:34:02
From: Leduc
Subject: Hi

I'm doing a report for my math class on the relation between pi and 
probability. I've been looking on the Internet for information, but
eveything I come across is way beyond my level. Could you send me some
info? I'm an 11th grader taking Advanced Functions.

Could you also send me info on the different ways to calculate pi.  
I'm doing a science project on this. I know so far that there is a way 
to calculate pi using arctangent, but I'm not sure how.

Claire

Date: 12/23/97 at 15:31:42
From: Doctor Rob
Subject: Re: Hi

Search the World Wide Web looking for "Buffon" together with "needle".  
I know it sounds crazy, but do it anyway. That will help you with 
part 1.

There is a lot of information on the many, many ways to calculate Pi.
You can try the following URL on the World Wide Web, and the pages 
linked to it:

   http://mathforum.org/dr.math/faq/faq.pi.html   

-Doctor Rob,  The Math Forum
 Check out our web site!  http://mathforum.org/dr.math/   

Date: 01/14/98 at 16:10:30
From: Leduc
Subject: Hi

Could you please tell me how probability is related to the normal
distribution curve or any other aspects of probability?

Thanks,
Claire

Date: 01/14/98 at 17:14:01
From: Doctor Rob
Subject: Re: Hi

Thanks for asking, Claire.

If you have an event with probability 1/2, like getting heads when
you flip a coin, and you count the number of successes after n trials,
for some positive integer n, you will find that the probability of
getting exactly k successes and m failures, where k + m = n, is given 
by

   b(k; n, 1/2) = 2^(-n)*n!/(k!*m!),

where r! = r*(r-1)*...*2*1 is "r factorial."

Define h = 2/Sqrt[n], and x(k) = (k - n/2)*h, and then it turns out 
that

   b(k; n, 1/2)
   ------------
    h*phi[x(k)]

approaches 1 as n and k grow without bound in such a way that
x(k)^3/Sqrt[n] -> 0.  (This means k stays "near" n/2).  Here

   phi(x) = e^(-x^2/2)/Sqrt(2*Pi),

the normal distribution curve. This can be generalized to a situation
where the probabilities of success and failure are not equal, as well.
An example is rolling a die, and getting a 3, whose probability of 
success is 1/6, and of failure 5/6. This is so famous a theorem that 
it has a name:  The DeMoivre-Laplace Limit Theorem.

This means that for k and n satisfying those conditions,

   phi[x(k)] ~=~ b(k; n, 1/2)*Sqrt[n]/2.

This approximation is surprisingly good, even for fairly small n's, 
like perhaps 50.

Thus you could say that the normal distribution curve can be used to
approximate the distribution of successes in coin-flipping with a 
large number of trials.

I hope that this is what you wanted.

-Doctor Rob,  The Math Forum
 Check out our web site!  http://mathforum.org/dr.math/