# Probability with Percents

A local population consist of 80% women and 20% men. If we select a sample of 20 people at random from this population, then how many woman can we expect to be selected in the sample?

Any random sample from the population will still be $80$ percent women and $20$ percent men. What's $80$ percent of $20$?
If X is any one person from the population, then all we know from the problem is. $$p(X = woman) = 0.8$$ so $$E(\sum_{i=1}^{20}\delta(X_i = woman)) = \sum_{i=1}^{20}E(\delta(X_i = woman))$$ $$= \sum_{i=1}^{20}p(X_i = woman) = \sum_{i=1}^{20}0.8 = 16$$