Calculate probability of selecting x people of certain gender from population

Say I have a population which is evenly split into men and women. If I'm randomly selecting a sample of y, how do I calculate the probability of selecting a specific split by gender?

Example:

Total population = 150 (75 men and 75 women)

Randomly select 20 people.

What is probability of selecting 0 men and 20 women? 1 man and 19 women? 2 men and 18 women? ... etc.

Furthermore - what is the formula to generalise this if it is an unevenly split population (e.g. there are 80 men and 70 women to begin with in the example above)?

1 Answer

Let your total population be $T = M + W$ where $M$ is the number of men and $W$ is the number of women.

Then there are $\binom{T}{20}$ ways of choosing $20$ people. If you want to know the probability of selecting $m \le \min\{20, M\}$ men and $w \le \min\{20,W\}$ women (where $m+w = 20$), then you have $\binom{M}{m}$ and $\binom{W}{w}$ ways of selecting the men and women, respectively. Therefore, your probability is

$$\frac{\binom{M}{m}\binom{W}{w}}{\binom{T}{20}} = \frac{\binom{M}{m}\binom{W}{w}}{\binom{M+W}{m+w}}$$