Hi, the question involves 3 parts.
1.) A family has 3 children. Event A= at least 1 girl, B= children of both genders. Show that A and B. are independent.
2.) If there are 2 children, use A, and B to show they are NOT dependent
3.) If there are 4 children use A, and B to proven they are NOT independent
It's mostly 3 that I'm not sure about, I've uploaded photos of my work
Thank you
You don't care about the order of the genders at all so there is no reason to distinguish between say bgb and bbg. So your population is given by {bbb,bbg,bgg,ggg}.
Assuming P(b)=P(g)=1/2. Your distribution is
Pr[{bbb,bbg,bgg,ggg}] = {1/8, 3/8, 3/8, 1/8}
Now A is {at least 1 girl} not {at most one girl} as you've written.
So Pr[A] = Pr[bbg]+Pr[bgg]+Pr[ggg] = 7/8 not 1/2 as it looks like you've calculated.
Pr[B] = Pr[bbg]+Pr[bgg] = 3/4
Go back and rework things. I think you have the general idea.