Details

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 The Hunger 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.