# 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 $20$?

\begin{align*} .80(20) = 16 \end{align*}

• this needs an "in expectation" somewhere. Certainly not every random sample will be 80 percent women and 20 percent men. – WetlabStudent Feb 28 '15 at 1:07

To show the answer using formal computation of expected values,

If X is any one person from the population, then all we know from the problem is. $$p(X = woman) = 0.8$$ so $$E(\sum_{i=1}^{20}\delta(X_i = woman)) = \sum_{i=1}^{20}E(\delta(X_i = woman))$$ $$= \sum_{i=1}^{20}p(X_i = woman) = \sum_{i=1}^{20}0.8 = 16$$