Poisson Distribution with Drawing Balls

Question

If two series of $100$ draws are made from a box at random. The box has $1000$ balls where there are $2$ black balls and the rest are all white. What is the probability that they both produce the same amount of black balls?

I want to use a Poisson Distribution here, however I don't know how to start.

Answer 1

If I understand the problem, the number $X$ of black balls drawn is distributed approximately as $POIS(.2)$ and (per the comment of @drnk) has exactly the hypergeometric distribution for 100 draws from a box with 2 black and 998 white balls.

Either way, the random variable $X$ can actually take only values 0, 1 and 2, and the answer is about 0.7.

Computations in R statistical software:

 i = 0:2;  pois.pdf = dpois(i, .2);  hypr.pdf = dhyper(i, 2, 998, 100)
 sum(pois.pdf^2); sum(hypr.pdf^2)
 ## 0.697401
 ## 0.6885172

The two distribution tables are as follows (noting that the approximate Poisson probabilities ignore values greater than 2 and do not add exactly to 1).

 cbind(i, pois.pdf, hypr.pdf)
      i   pois.pdf   hypr.pdf
  ##  0 0.81873075 0.80990991
  ##  1 0.16374615 0.18018018
  ##  2 0.01637462 0.00990991

 sum(pois.pdf)
 ## 0.9988515  # not exactly 1