Probability of choosing two girls from a group

Question

A group contain $9$ boys and $3$ girls.

For a trip, we choose randomly a group of $4$ from the group above.

What's the probability that half of the trip group will be girls?

I would like to know how to solve it without using combinatorical approach.

My solution : $p=\frac{(9 ncr 2)(3 ncr 2) }{(12 ncr 4)} =\frac{12}{55} $.

Thank you.

Answer 1

A probability approach can be used, but note that a multiplication factor will be needed.

$P(GGBB)\; in\;that\;order\; = \frac3{12}\frac2{11}\frac9{10}\frac89= \frac{2}{55}$

Since there can be $\frac{4!}{2!2!}= 6$ possible orders, ans $= \frac{12}{55}$

Answer 2

You could obtain the same answer by drawing a 4 - generation tree diagram with sampling-without-replacement probabilities on all the branches. However the combinatorial approach is quicker and easier.