In a set of N individuals X% have some characteristic.
In a random subset of M individuals of this set, we observe that Y% have the characteristic. Note that M is not necessarily very small compared to N.
1) What is the probability P(N, M, X, Z) that | Y% - X% | <= Z% ?
2) What is the probability P'(N, M, X, Z) that Y% - X% > Z% ?
Example: There are 200 inhabitants in a village. 12% have blue eyes. In a random sample of 20 inhabitants, what is the probability that, among them, the proportion having blue eyes is between 10% and 14% (i.e. 12% +/- 2%)? Strictly greater than 14%?