probability of occurrence in small sample of population when know occurrence in population

Question

If 2% of an overall population of 5000 has a certain characteristic, what is the probability that 2 or more of a subset of 10 drawn from the population will have this characteristic? Is there a formula or common test such as a Z-score to figure this out?

Answer 1

Note that P(2 or more have characteristic) = 1 - P(0 or 1 have characteristic)  

 

P(0 have characteristic) = 0.98¹⁰ = 0.8171 

 

since all 10 of the individuals must not have the characteristic 

 

1 individual out of the 10 can have the characteristic in 10 different ways (it can be any 1 out of the 10 individuals) times the probability that 9 in the sample do not have the characteristic and 1 does, which is 

 

P(1 has characteristic) = 10*0.98⁹*0.02¹ = 0.1667  

 

So P(2 or more) = 1 - 0.8171 - 0.1667 = 0.0162 

 

There are some situations where a binomial distribution like this can be approximated by a Normal distribution and you would end up using a Z-score, but in general the number of samples n must be large enough so that both n*p and 

n*(1-p) are both greater than 10.  In this case n = 10 and p = 0.02 so n*p = 0.2 so you must resort to using normal probabilistic methods.