Sampling Without Replacement

Photo by Dennis van Zuijlekom

Sampling Without Replacement is a probability question that is an example of using the Hypergeometric Distribution.

If the sample size was great compared to the population size, then we could use Binomial distribution to approximate.

Equation and Terms

Hypergeometric probability

Hypergeometric probability


Remember that C(a / b) = a! / (b! * (a-b)!)

Population Size: N =

Sample Size: n =

Occurrences in sample: r = 0

Occurrences in population: d =



5 books are damaged out of a set of 15. What is the probability of selecting 3 undamaged books when you pull without replacing?

First, compare the sample size to the population.

We see that the sample size (3) is small compared to the over all population size (15). Thus, we will use the hypergeometric distribution.

Next we identify the terms:

Hypergeometric probability

Hypergeometric probability

Population Size: N = 15

Sample Size: n = 3

Occurrences in sample: r = 0

Occurrences in population: d = 5

P = C(5/0) * C(10/3) / C(15/3)

C(a/b) = a! / (b! * (a-b)!)

P =  0.2637




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