Solving the Boy Girl Paradox Probability for Random Families

[Verdict]

Homework Statement

Suppose you choose a random family with two children. One of them is a
girl. What is the probability that both children are girls?
Derive your answer by explicitly constructing an outcome space and a probability
measure, and naming the relevant events in terms of this outcome
space. You may assume that the probability that a child is male or female is
1/2, independently of the gender of another child.

The Attempt at a Solution

Alright, so my question is fairly straightforward; what situation am I dealing with here? If I can trust wikipedia, there are two common ways of phrasing the issue, namely

From all families with two children, at least one of whom is a boy, a family is chosen at random. This would yield the answer of 1/3.

From all families with two children, one child is selected at random, and the sex of that child is specified. This would yield an answer of 1/2.

Now, I understand the reasoning behind both of them, but I can't decide in which category my version of the question falls. It seems to have characteristics of both of them. A random family of two children is chosen, so that sounds like the 1/2 case. But a child is not identified or anything, so that makes it sound more like 1.

Can anyone help me out?

 

[cepheid]

Label the children with numbers (to distinguish them) and letters (to specify gender). Then if my outcome space is:

g1 g2
g1 b2
b1 g2
b1 b2

and we eliminate the last outcome, because we know one of them is a girl, and if we consider all outcomes equally likely, so that our probability measure is (number of desired outcomes)/(number of possible outcomes), then I would say the answer is 1/3.

So I think your problem falls into category 1. The problem just says, "at least one of them is a girl", it doesn't specify which one. If the problem said, "the first child is a girl" or "the second child is a girl", then obviously we'd eliminate all outcomes that didn't have g1 (or g2, as the case may be), leaving only two possible outcomes. The answer would then be 1/2. This is what you called category 2.

 

[Ray Vickson]

Homework Statement

Suppose you choose a random family with two children. One of them is a
girl. What is the probability that both children are girls?
Derive your answer by explicitly constructing an outcome space and a probability
measure, and naming the relevant events in terms of this outcome
space. You may assume that the probability that a child is male or female is
1/2, independently of the gender of another child.

The Attempt at a Solution

Alright, so my question is fairly straightforward; what situation am I dealing with here? If I can trust wikipedia, there are two common ways of phrasing the issue, namely

From all families with two children, at least one of whom is a boy, a family is chosen at random. This would yield the answer of 1/3.

From all families with two children, one child is selected at random, and the sex of that child is specified. This would yield an answer of 1/2.

Now, I understand the reasoning behind both of them, but I can't decide in which category my version of the question falls. It seems to have characteristics of both of them. A random family of two children is chosen, so that sounds like the 1/2 case. But a child is not identified or anything, so that makes it sound more like 1.

Can anyone help me out?

Even more paradoxical: if you say that the oldest child is a girl, you get an answer 1/2. If you say the youngest child is a girl, you get an answer 1/2.

Sometimes it helps to think of equivalent coin-tossing games. So, if we toss a coin twice and say "there is at least one H", the probability is 1/3 that the other toss is also H. However, if you say that the first (or second) toss is H, the probability is 1/2 that the other is also H.

 

[Verdict]

Hmm alright, I was also opting for the 1/3 case, and indeed mainly because the 'category 2' asks for you to identify a child of a certain sex. Nothing is said about which one in the question, I agree with you on that.

 

[tiny-tim]

Hi Verdict!

Suppose you choose a random family with two children. One of them is a girl. What is the probability that both children are girls?
…
From all families with two children, one child is selected at random, and the sex of that child is specified. This would yield an answer of 1/2.

I honestly don't see how you get "one child is selected at random" from "One of them is a
girl"

Does it say "One of them chosen at random is a girl"? (or "the first one is a girl")

Nope!

You'd have to read the word "random" twice into the question (or the word "first") … stop seeing things that aren't there!

 

[Verdict]

Hi Verdict! I honestly don't see how you get "one child is selected at random" from "One of them is a
girl"

Does it say "One of them chosen at random is a girl"? (or "the first one is a girl")

Nope!

You'd have to read the word "random" twice into the question (or the word "first") … stop seeing things that aren't there!

No no, you are right, but that is not why I was confused. I was confused because in my question, a family of two children is chosen at random. In the first case of the two versions on wikipedia, a random family of two children of which at least one is a girl is chosen. That's why I wasn't sure. I agree that it did not say that the one chosen at random is a girl, I just wasn't sure.

That, and the question is much harder (in my opinion) if you have to recognize that it is case 2. If not, it is rather straightforward.

 

[tiny-tim]

hmm …

i can't help feeling that if you hadn't read wikipedia, you wouldn't have been confused! ​

 

[haruspex]

Suppose you choose a random family with two children. One of them is a girl.

No marks to the problem setter! This is a classic blunder in posing this classic problem. In common parlance, what is a reasonable way to interpret "One of them is a girl"? If I say "I have two daughters; one of them is a doctor", the reasonable listener will presume the other is not. OK, we can dismiss that because of the way the question continues, but it still leaves open the question of how the 'one' was chosen. It reads as though I, the one who chose the family, happen to notice that one of them is a girl, and have not yet checked the gender of the other child.
The wording ought to be something like "at least one is a girl", or maybe better, "they are not both boys".

 

[Borek]

Suppose you choose a random family with two children. One of them is a girl.

You either choose a random family from all families that have two children, or random family from all families that have two children of which at least one is a girl. These are different sets of families.

 

[Jdscalise2000]

Isnt there another possible combination to factor? We eliminate gg because there is already 1 boy. So wouldn't the possibilities be: bg (older boy younger gir), gb (older girl younger boy), bb ( older boy younger boy) and then bb again? Meaning don't we factor that the older boy could be the younger and the younger could be older?