A couple has two children, one of whom is a girl.

Hi,
Do not get into kids as B or G,try with another example
say there are two different flavours of icecream - V and C..
50% chances that one can pick either and eat..
we know he ate two and one was surely a V.. what is the probabilty that he had two vanillas, V..
he could have had VC or CV or VV

well neither we need to apply the twin or age or any other theory. lets put it this way:

I am told that my neighbour has two child. one is girl another may be boy or girl. I went to my neighbour's house. here are the total case:

1. a boy opens the door (then other child as we already know is girl) hence - BG.
2. a girl opens the door. (then other child could be boy or girl )- hence (GB or GG)

This is why we have these three cases.

GG is favorable out of three. so 1/3.

Thanks chetan. Your assistance helps! Kudos

Hi Bunuel,
If we consider above method, does it mean that the given probability of either boy or girl being 50% is not required to solve the problem?
And also can you please tell, the given probability translate into which of the below equations?:-
P(B U G)= 0.5
or
P(B) = P(G)= 0.5

*U = Union

VERITAS PREP OFFICIAL SOLUTION:

While every bone in your body wants to say 1/2, that is not correct! This is tricky conditional probability and considering the different possibilities is the best way to get it correct. If the couple has 2 children then there are 4 possibilities: BB, BG, GB, GG.

If one of the children is a girl then only 3 possibilities remain: BG, GB, GG. Of those 3, only one is favorable to create 2 daughters (GG) so the answer is 1/3. Answer is (C )

OFFICIAL SOLUTION
Solution: While every bone in your body wants to say 1/2, that is not correct! This is tricky conditional probability and considering the different possibilities is the best way to get it correct. If the couple has 2 children then there are 4 possibilities: BB, BG, GB, GG.
If one of the children is a girl then only 3 possibilities remain: BG, GB, GG. Of those 3, only one is favorable to create 2 daughters (GG) so the answer is 1/3. Answer is (C )

As the order isn't given here, we can't remove BG form the same set. All we may remove is BB, Boy and a Boy as this is an irrelevant answer option for us here.
As we already know that one of the Child is girl.

Bunuel --- Need your help to shine some more light on this, if possible. (I know this post is very old)

How can we use the probability approach in this question?
I understand that one of the children is already a girl. The probability that the other child is a girl is 1/2.
How does that line of reasoning completely fail here?

Thanks in advance. I am trying to understand the probability approach better here but am failing.

The answer is 1/3.

Let's consider the case that probability is 1/2, if one of the two child is a girl (as many argued). Please check what is the probability if the question has been that the eldest child is a girl, than also it would be 1/2 in your case. The two probabilities cannot be same.

Probability of having DD =
DD/DB+BB+DD
=1/3

Posted from my mobile device

For people who want to think that birth order doesn't matter, that BG is no different than GB, the the situation would be:

BB
BG/GB
GG

and you'd have to then acknowledge that the BG/GB scenario is twice as likely as BB or GG, and that BB and GG are equally likely.

So if probability of GG or BB are each X, then probability of BG/GB is 2X.

Being told that one child is a girl (not only one), then the probability of GG would be X/(X+2X)= X/3X = 1/3

Of course, if you examine this approach you would realize that you actually ARE considering birth order.

Posted from my mobile device

Here is a simple approach for conditional probability problems:

Recall that in conditional probability, the entire probability space has been restricted.
What does this mean?
Consider the question.
The full probability space would be (for two children):
Boy, Boy
Boy, Girl
Girl, Boy
Girl, Girl


Now, we know that there is one girl - therefore we can remove Boy, Boy our space is now:

Girl, Boy
Boy, Girl
Girl, Girl

To get our desired (girl,girl) we have 1 case out of our restricted 3 cases. So 1/3.

Note that the order does matter - although the end result is the same (one girl and one boy) the event is different. It's like flipping a coin - heads tails and tails heads are a different sequence, and hence different events