How to find this probability with bags of balls?

Question

I have two bags with different color of balls in it. In bag 1 there are 1 white ball and 3 black balls and in bag 2 there are 1 white ball and 2 black balls. I wanna know what is the probability for me getting one black ball and one white ball if I choose one of the bag randomly and grab two balls out from it at once. Hence, what would be the probability for me taking out a black ball and a white ball from the bag 2 instead of bag 1?

my attempt: using binomial to test out the chance but I have gotten some really complicated answer. Would anyone be able to help me with it? Any help would be appreciated

Answer 1

$$P(\text{one black, one white})=P(\text{one black, one white}|\text{Bag 1})\cdot P(\text{Bag 1}) + P(\text{one black, one white}|\text{Bag 2})\cdot P(\text{Bag 2}).$$

$$ P(\text{one black, one white})= \frac{\pmatrix{3\\1} }{\pmatrix{ 4\\2}}\cdot \frac{1}{2} + \frac{\pmatrix{2\\1} }{\pmatrix{ 3\\2}}\cdot \frac{1}{2}$$

$$ P(\text{one black, one white})=\frac{1}{4} + \frac{1}{3}=\frac{7}{12}.$$

$$ P(\text{one black, one white from Bag 2})=\frac{\pmatrix{2\\1} }{\pmatrix{ 3\\2}}\cdot \frac{1}{2}=\frac{1}{3}.$$