Banana Stickers

Date: 05/29/97 at 15:11:52
From: nathan & jim
Subject: Banana - probability 

You collect banana stickers and each banana sticker has a different 
letter on it (a-z). If you collect n {n: n > 26} banana stickers, what 
is the probability that you'll have all 26 letters?

Date: 06/01/97 at 20:17:23
From: Doctor Anthony
Subject: Re: Banana - probability 

We can tackle this by what is known as the urn model.  You have 26 
urns and n balls, and the balls are distributed at random among the 26 
urns.  What is the probability that there are no empty urns?

The number of ways of distributing n numbered balls into 26 numbered 
boxes such that no box is empty is given by T(n,26).  The total 
number of ways of distributing the balls without restriction is 
26^n, so the required probability is

     T(n,26)
    --------
       26^n 

T(n,26) will be the coefficient of x^n/n! in the expansion of

  [e^x - 1]^26  = e^(26x) - C(26,1)e^(25x) + C(26,2)e^(24x) - ..

The term in x^n/n! will be

  (x^n/n!).[26^n - C(26,1)25^n + C(26,2)24^n - .....]

and the term in square brackets is  T(n,26)

This will be a very lengthy calculation but it is necessary to 
ensure that our probability is calculated from equiprobabe events.

You might also want to take a look at this related item from our 
archive:

  Collecting a Set of Coupons
  http://mathforum.org/library/drmath/view/56657.html

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