- Background Information:
I am studying discrete math, and I have come across an example that my professor used the class. She forgot to explain the second part of the question, I need clarification understanding what to do.
- Original Question:
What is the conditional probability that a family of two children has two boys, given they have at least one boy? Assume that each of the possibilities BB, BG, GB, GG is equally likely, where B represents boy and G represents girl.
In the above example, are the events E, that a family with two children has two boys, and F, that a family with two children has at least one boy, independent?
- My Work:
The total possibility is: {BB, BG, GB, GG}
"What is the conditional probability that a family of two children has two boys, given they have at least one boy? "
Given that they have at least one boy then our total possibility is {BB, BG, GB}.
P(BB) = 1/3 where P stands for probability
I also know that independence means P(A ∩ B) = P(A)P(B)
- My Question:
I understand what independence means, but how am I supposed to show it here?
"In the above example, are the events E, that a family with two children has two boys, and F, that a family with two children has at least one boy, independent?"
No, they are not at all independent, since event $E$ implies event $F$.
That is, any family with two boys and no other children is automatically a family that has two children with at least one boy.
Indeed, we have $P(E \cap F) \not = P(E)\cdot P(F)$, since:
$P(E \cap F) = P(E) \not = P(E) \cdot P(F)$