What is the Probability of Mr. and Mrs. X Having Two Boys?

[retupmoc]

In my stat mech book i seem to have a problem with the answer to this question which seems really straight forward;

Q/ Mr and Mrs X have 2 children. If they tell you that at least one of them is a boy, but you don't know the gender of the other child, what is the probability that they have two boys?

A/ 1 in 3. There are four 'microstates': - BB BG GB and GG. By telling you that at least one of the children is a boy Mr and Mrs X have only eliminated the GG microstate. The point is that there are two microstates (BG and GB) in the 'macrostate' of having a boy and a girl. Children are distinguishable particles

My problem with this, is that since we are told that at least one of the children is a boy which leaves only the possibility that the other child is either a boy or a girl, hence a 1 in 2 chance. Why does the order BG or GB matter in such a scenario?

 

[Tide]

BG and GB represent two distinct ways in which you can arrive at a boy and a girl. It might help to think of one as the older child and the other as the younger. In calculating probability you must account for all possible "outcomes."

 

[werner heisenberg]

I cannot see the answer of the question. The two arguments posted by retupmoc seem to be correct. I would say this is a paradox

 

[Dale]

The reason that it matters is that they did not tell you if it was the older or the younger child that was a boy. So your 1/2 reasoning is correct if they say "the older child is a boy what is the probability of having 2 boys" or if they say "the younger child is a boy what is the probability of having 2 boys". In each of those cases the probability that the other child is a boy is 50%.

However, the problem at hand states only "at least one of them is a boy". You do not know which one is a boy. So if the first one is a boy then there is a 50% chance that the second one is a boy and a 50% chance that it is a girl. But if the first one is a girl, then the second one must be a boy. This leads to 3 possible scenarios that fit the description "at least one of them is a boy". Out of those 3 only 1 corresponds to two boys. So the probability is 1/3. As the answer mentions, children are distinguishible particles, they are distinguishable by age.

-Hope that helps
Dale

 

[HallsofIvy]

You are correct that, not knowing anything a-priori, the 4 "equally likely" events (why, oh, why would anyone say "micro-states"??) are GG, GB, BG, BB where, exactly as you say, GB and BG have to be treated as separate because "Children are distinguishable particles" (Oh, I see- this is a physics course! But even so--. Forgive me if I refuse to speak of children as "particles" or having children as "microstates"!). As Tide said, it might help you to think of the first as being the older child: BG means older boy, younger girl, GB means older girl, younger boy.

Knowing that "at least one of the children is a boy" removes the "GG" possiblity leaving BB, GB, BG as the equally likely events. There are 3 of them and exactly one corresponds to two boys- What is the probability?

If we were told "the oldest child is a boy", that would remove both
"GG" and "GB" from the mix: leaving BB and BG as the possiblities. Then the probability of two boys would be 1/2- but that's a completely different problem.

 

[Dale]

Forgive me if I refuse to speak of children as "particles".

I don't know, I kind of like the idea. I have only been collecting data for about 5 years, but so far I have a great deal of impirical evidence that indicates strong similarity between particles and children.

1) They both seem to have a propensity for pretty mindless high-energy collisions.

2) The collisions of the particles seems largely governed by the surrounding electromagnetic fields generated by accelerators while the collisions of children seems largely governed by the surrounding breakable-fields and valuable-fields generated by household objects.

3) The energy of the collisions is amazingly high considering the relatively low masses involved.

4) The path of a particle is influenced by the stress-energy tensor of space much like the path of a child is influenced by the stress-energy levels of the parent.

5) Much like muon decay, the "are we there yet" experiment provides conclusive evidence of time dilation.

6) They have only two possible energy levels, awake and asleep. That and the fact that you can only have an integer number of children indicate that they are a quantum phenomenon.

7) As with most quantum phenomena their behaviour can only be determined in a probabilistic sense.

Of course, the mental and emotional responses involved in child-child and child-parent interactions seem impossible to fit into the particle model of children. I think the best course therefore is to introduce the concept of particle-person duality.

-Dale

PS Sorry about getting so far off topic, but I couldn't resist once I got the idea.