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# A Quick Question On Probability watch

1. I was wondering if TSR masses could help settle a debate I'm having with a friend.

We have two problems.

The first problem:
A mother has been told she is giving birth to twins. She wants to have a daughter, but does not mind what sex her other twin is. What is the
probability that at least one of the twins is a girl? Assume that each baby has an equal chance of being either a boy or a girl.

Now, I reckon she has four possible outcomes of having twins (assuming a 50% chance of either a boy or girl): GG; GB; BG; and BB; with a probability of 0.25 each. The outcomes that give at least one girl are GG, GB and BG, giving a probability of 0.75 that at least one twin is a girl.

The second problem:
The doctor asks her whether she wants to find out the sex of her twins.
She tells the doctor that she only wants to know whether she is having a girl, and the doctor confirms that she is. The doctor does not reveal the sex of the other child. Given that she knows one of the twins is a girl, what is the probability of the other one also being a girl? Again, assume that each baby has an equal chance of being either a boy or a girl.

This is causing the debate! I say that if she knows that at least one of
the twins is a girl, she has three possible outcomes: GG, GB and BG. Only one of these allows for the possibility of the other one also being a girl: GG. Therefore she has a 1/3 chance of both twins being a girl if she knows that one twin is definitely a girl.

Could anyone possible check this and let me know if these answers are correct?
Thanks very much!
3. Cheers!

For the sake of completion, this is what my mate's saying:

No, sorry you're wrong (and right). There are three possible outcomes but they are not all equally likely.

I'll try explaining it a different way.

Given the original caveat (that a child is equally likely to be a boy or a girl, then the total for all probabilities that involve child 1 being a boy must be 50%. Likewise for a girl - 50%. There is only one outcome that involves child 1 being a boy (as there must be a girl) therefore that must have a probability of 50%. If child 1 is a girl then there is a 50% chance of child 2 being a girl - in other words 50% of 50% (or 25%) for it being girl-girl. The same for girl-boy (25%).

Boy (50%) - Girl (100% of 50%) = 50%

Girl (50%) - Boy (50% of 50%) = 25%

Girl (50%) - Girl (50% of 50%) = 25%

The whole point is that the 3 pairings are not equally likely even though it may appear so at first glance.

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