+0

# Probability with Combinations

+1
48
3
+485

The Grammar club has 20 members: 10 boys and 10 girls. A 4-person committee is chosen at random. What is the probability that the committee has at least 1 boy and at least 1 girl?

Apr 3, 2021

#1
+1

There is a probability of 1/2 of choosing a boy, then a 10/19 probability of choosing a girl.  Then the other two people can be anyone, so the probability is 2*1/2*10/19 = 10/19.

Apr 3, 2021
#2
+485
+1

That was incorrect, but I have one more try.

RiemannIntegralzzz  Apr 3, 2021
#3
+485
+1

Actually, I figured it out myself:

You can use complementary counting/combinations here:

The number of ways to choose a committee of all boys or all girls is $2*C(10,4)$(2 for the girls and boys)$=420$. The total number of committees is $C(20, 4)=4845$. So the answer is $1-\frac{420}{4845}=\frac{4425}{4845}=\boxed{\frac{295}{323}}$

Apr 3, 2021