How to determine conf interval about expected number based on probability?

From: Me (hlpme2004_at_hotmail.com)
Date: 08/29/04 

Date: 29 Aug 2004 14:18:55 -0700

As an academic curiosity, in a lottery game, given P[A] --
the probability of matching A of B numbers chosen from C
numbers -- the expected number of winners is:

E[A] = n * P[A]

where n is the number tickets sold.

Suppose we want to be X% confident that the number of winners
will be E[A] +/- delta.

How do we determine delta?

I believe that:

delta = z * sd

where sd is the std dev, and z is the t-value for the X%
confidence level. For example, z is 1.0, 1.645, 1.96 and
2.576 for 68.3%, 90%, 95% and 99% confidence.

So I believe my question becomes: how do we determine the
std dev?

Consider a lottery game where we select 5 of 47 numbers.
Then, P[3] is 8,610 / 1,533,939 [1]. If 3,067,878 tickets
are sold, E[3] = 17,220.

We want to be 95% confident that the number of winners will
be 17,220 +/- delta.

What is delta? Please show how delta is derived as a
function of E[3], P[3], n and/or the 95% confidence level
(t-value 1.645).

-----

[1] P[3] = C(5,3) * C(42,2) / C(47,5)
         = 10 * 861 / 1,533,939
         = 8610 / 1,533,939

    C(5,3) = number of groups of 3 matching numbers in 5
    C(42,2) = number of ways to select 2 from 42
              non-matching numbers
    C(47,5) = number of ways to select 5 numbers from 47