Three students are chosen at random from a class of ten girls and twelve boys. What is the probability that at least one is a boy given at least one is a girl.
Would this be an independent or dependent event? I believe is dependent since they are not replacing their picks thus the pool gets smaller.
If that is the case then would it be $P(A\cap B)$= P(A) * P(B|A)? I'm thinking probability of ate least one boy is P(A)=$\frac{6}{11}$, and probability of at least one girl is P(B)= $\frac{5}{11}$.
Best Answer
Let $X$ be the number of boys chosen among the three. You have $X\sim\text{Hypergeometric}(22,12,3)$. Then:
$$P(\text{at least one girl})=P(X=0)+P(X=1)+P(X=2)=\frac{6}{77}+\frac{27}{77}+\frac{33}{77}=\frac{66}{77}$$
$$P(\text{at least one boy and at least one girl})=P(X=1)+P(X=2)=\frac{60}{77}$$
$$P(\text{at least one boy}|\text{at least one girl})=\frac{P(\text{at least one girl and at least one boy})}{(\text{at least one girl})}=\frac{10}{11}$$