[NMSU, N. Vogt]

The probability that any one dart will fall into a particlar slot is p = 1 / N.

The probability that n darts fall into a particular slot can be calculated as a series of Bernouli trials.

For the case n = 0 (the slot is empty),

We can also show that

and for the case x = -1,

and by combining these two results we see that, as N goes to infinity,

If we perform our experiment, we can count Z as the number of slots without any darts. Then