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First argument:

Suppose we throw 1 red billiard ball on the table and measure
how far it goes on the scale from 0 to 1,
call this value *x*,
then throw *n* balls,
what is the distribution of the number
of balls to the left of the red ball?

Now what is *x* distribution?
It is
*Uniform*(0,1), and the overall
distribution of the number of successes is
the sum for all possible *x*'s:

Second way:

Suppose I throw all the balls down first, and choose which is
to be the red one, then the probability that the red one
has *k* to the left of it is:
.
So we have:

Which tells us that:

This constant actually has a special
name it is
*B*(*k*+1,*n*-*k*+1) and it comes in
for the following important
density function, whose support is
[0,1].

*Susan Holmes*

*1998-12-07*