In probability theory, the "occupancy problem" considers throwing stones into buckets. If each stone has equal probability of entering any bucket, what is the probability there are empty buckets?

Given buckets and stones, the probability that there are no empty buckets is .

More generally, the probability that of buckets are empty after throwing stones is .

Example 1: if you roll a fair die 10 times, what is the probability that you see all six sides?

Example 2: In her book TheHunger Games: Catching Fire by Suzanne Collins, after 74 games one male and one female victor are selected from 12 districts (, ). Is it reasonable to assume each district has a female and a male victor?

Displayed in blue for is the expected number of stones to throw to get at least one stone in each bucket , where is the harmonic number . This problem is also known as the coupon collector's problem with different kinds of coupons. Many coupons are distributed randomly; the above relation gives the probability of having the complete set after purchases.

Reference

[1] W. Feller, An Introduction to Probability Theory and Its Applications, Vol. I, 3rd ed., New York: John Wiley & Sons, 1968.