# Conditional Probabilities

## Homework Statement

A couple has two children. What is the probability that both are girls given that the oldest is a girl? What is the probability that both are girls given that one of them is a girl?

DO NOT assume that female births and male births are equally likely. Assume identical twins are necessarily the same sex.

Write your answer in terms of the probability of a female birth and the probability of identical twins (assume that identical twins are just as likely to be female as the overall female probability).

Bayes Theorem?

## The Attempt at a Solution

All right the problem as stated in the first paragraph without extra stipulations is easy, but I am unsure about how to complete solution now. Considering the question "What is the probability that both are girls given that the oldest is a girl" , if we let A = event that both are girls and B = oldest is a girl. Then by Bayes,

P(A|B) = P(B|A)P(A) / P(B). But is this the correct way to go? Isn't P(B) just P(female birth)?

lanedance
Homework Helper
is there anything that says the sex of one child is dependent on another for a given couple?

if they are independent events then things simplify

Dick
Homework Helper
is there anything that says the sex of one child is dependent on another for a given couple?

if they are independent events then things simplify

They are dependent if they are identical twins. There are two cases here. They are identical or they are not identical. Only the second is independent.

I'm afraid to assume independence of probability here (in general I've been told it's a bad idea, but it seems a bit strong). By the way independence is not a property of events, but rather probabilities (we can have disjoint events, but these coincide with independent probabilities only in a specific case). I know what you meant and I probably have been confused about this distinction in the past myself.

lanedance
Homework Helper
nice point Dick, that does introduce the dependence

as there is not a heap of events, i would start by drawing a probability tree, first branch is whether or not they have identical twins, 2nd barch is sex of 1st child (ends heres for twins branch) 3rd branch is sex of the 2nd.

you know they don't have identical twins after having a single child as they only have 2 children.

from there it should be easy to read off probabilities & relate to bayes, if needed

Thanks lanedance, I drew out the tree as you suggested. I'm a little shaky on the rules of adding and multiplying probabilities (not thinking clearly at the moment), but I'll try to see what I come up with.

lanedance
Homework Helper
tips -
- put the probabilty of the outcome of of each event on each branch
- the probabilty of the final outcome will be all teh probabilties on each branch l;eading to it muliplied
- check the sum of all the probabilties of final outcomes sums to 1
- label each outcome as one of (MM, MF, FM, FF)

now to get the total porbabilty of FF, add the probabilties of all the FF outcomes. This should give you all the probabilties you need to use Bayes theorem

note also if A = event that both are girls and B = oldest is a girl
then P(B|A) = 1

Ah, so I don't really have to use Bayes? That was extremely useful, and I ended up with pq + (1-q)pp, where p = probability of female birth and q = probability of identical twins. Hopefully that's right. Thanks.

lanedance
Homework Helper
not sure how you have done it (might want to check that), but the use of Bayes is somewhat implicit
for example is

note also if A = event that both are girls and B = oldest is a girl
then you are effectively using Bayes with

P(A|B) = P(B|A)P(A) / P(B) = 1.P(A) / P(B) = P(A) / P(B) in this case as every case with 2 girls has the oldest a girl

so
P(A) = sum porbs of all cases with FF
P(B) = sum probs of all cases with Fx, where x is anything

then for this case
P(A|B) = P(A)/P(B)

Doh, I calculated all FF possibilities and thought I was finished with the problem. Sorry. Yes so I just worked out the other case P(both are girls | one is a girl) to be essentially the same, I think. Divide all FF possibilities over sum of final outcomes with at least one F. I hope this makes sense.

lanedance
Homework Helper
yeah i think that makes sense - so writing it out

P(boths girls, given oldest girl) = P(FF)/(P(FF) + P(FM))

P(boths girls, given 1 girl) = P(FF)/(P(FF) + P(FM)+P(MF))