Coverage Probability with the Occupancy Problem

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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?

Contributed by: Aaron Becker (January 2014)
Open content licensed under CC BY-NC-SA



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.


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

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