Random selection of a committee Hello and thanks for looking at my questions.
I'm sure this is a very simple question but I just cant seem to understand it
A committee of three is selected from a pool of five individuals : two females (A and B) and three males (C,D,E)
If the committee is selected at random, what is the probability that contains both females?
(i figured for this one it would be P=(2/5)*(1/4) because the first one selected has a 2/5 chance of being a women, and the second one has a 1/4 chance but I was told that was not correct)
and
If instead, the committee is selected by randomly selecting one of the two females, one of the three males, and then one of the remaining three individuals, what is the probability it contains both females
 A: This was wrong because you implicitly assumed that the two females will be choosen in the first two places. As, you point in the last three lines, this can be done also in a different order.
Using combinations (or even the hypergeometric distribution if you are familiar with it) you can approach it as follows


*

*The possible ways to choose the 3-person committee are $$\dbinom{5}{3}$$

*The favourable ways to choose the 3-person committee, i.e. the ways in which 2 females and 1 male are choosen are $$\dbinom{2}{2}\dbinom{3}{1}$$ Thus, by the classical definition (favourable over possible) we have that the requested probability is equal to $$\frac{\dbinom{2}{2}\dbinom{3}{1}}{\dbinom{5}{3}}=\large\frac{\frac{2!}{2!0!}\cdot\frac{3!}{2!1!}}{\frac{5!}{3!2!}}=\frac {3}{10}$$



Using the hypergeometric distribution (by denoting with $X$ the number of females in the committee), your paramaters would be $N=5$, $K=2$, $n=3$ (following Wikipedia notation) and you would be asking the probability $P(X=2)$.
A: For your first question the answer of Stefanos is excellent. For your second question:
If one female and one male have been chosen then there are $3$ persons left and exactly one of them is female. So the chance that a female will be chosen is $\frac{1}{3}$, and choosing a female in this situation is the 'same thing' as ending up with a committee that contains both females.
