1. ## Poisson Approximation Problem

A box contains 1000 balls, of which 2 are black and the rest are white.

a) which of the following is most likely to occur in 1000 draws with replacement from the box?

-fewer than 2 black balls
-exactly 2 black balls
-more than 2 black balls

b) if two series of 1000 draws are made at random from this box, what, approximately, is the chance that they produce the same number of black balls?

2. Originally Posted by Yan
A box contains 1000 balls, of which 2 are black and the rest are white.

a) which of the following is most likely to occur in 1000 draws with replacement from the box?

b) if two series of 1000 draws are made at random from this box, what, approximately, is the chance that they produce the same number of black balls?
Question a) is incomplete.

The number of black balls in a sample of $1000$ draws with replacement has a binomial distribution $B(1000,1/500)$ , which may be approximated by a Poisson distribution $P(2)$ .

The answer to b) is the sum:

$p\approx \sum_{i=0}^{\infty} [p(i,2)]^2$

where $p(i,\lambda)$ denotes the Poisson probability of outcome $i$, with mean $\lambda$.

The sum will for all practical purposes have converged by the time you reach $i=5$ or so.

CB

3. Original question updated for the poster.

CB