Conditional Probability of children

[succubus]

This is probably relatively easy, but I'm still a bit confused...

The question:

A family has j children with probability p1 = .1, p2 = .25, p3 = .35, p4 = .3.
A child is randomly chosen. Given this child is the eldest in the family, find the conditional probability that
a) Family has 1 child
b) Family has 4 children

A is easy (1), so let's move to b.

I have the following setup

E = event child is eldest
F = event family has 4 children

P(E) = (1*.1) + (.5 * .25) + (.3333 * .35) + (.25 * .3)

And

P(F) = .3

So the set up should be P(F|E) = .25*P(E)/P(E) ??

The events are independent correct? So I probably shouldn't include the ratio of how many children there are to choose from multiplied by the probability??


I'm kind of anxious and worried. Got 3 tests to study for an am sick. So I apologize if it's too easy :)

 

[Dick]

These problems tend to make my head hurt. And hence give wrong answers. But isn't the pool of children to select from only the oldest members of each family, given your given? If a) then reads 'at least one child', then sure. Probability 1. If a) reads 'exactly one child' isn't it just .1. Likewise for b) isn't it just .3? I may be more confused than enlightening here. Perhaps I don't understand the problem.

 

[succubus]

I think I figured it out. I think it's the probability of having 4 children (.3) times 1/4 divided by P(E) where
P(F|E) = .25 * .30
P(E) = 1*.1 + 1/2*.25 + 1/3*.35 + 1/4*.3


P(F|E) / P(E) ?

Does this seem right?