I'm sorry if I sounded offensive, and I indeed was, I was tired when I wrote my last comment
and my words didn't reflect my tought. I'd like to elucidate this problem once and for all, I really
do believe we can both agree on a conclusion.
I'd like you to notice that your new table of probabilities imply that I have a chance of 1/6 to guess the gender setup of a
family of 2 children. I don't think that makes sense either to you.
Let's make the experiment a bit clearer :
- We gather a number of families who have 2 childs.
- For each family, we announce the gender of one of the child, but we don't know wich one.
- We are then asked to guess the sex of the other child.
At this point, you believe that the probability to guess right is 1/2, and that the 2/3
probability doesn't make any sense. My claim is that you fall in the Monty Hall problem trap,
wich is very counter-intuitive and doesn't seem to make sense at first.
But here is some clarification of the problem :
- What we are really asked is to guess the _gender_ setup of the family. So we need to establish
the universe of possible family setups before answering. What are they ?
Even if we don't care about the order, we must acknowledge that there are 2 childs in
the family, so there must be a first child, and a second child.
Setup 1 : both childs are boys : M/M
Setup 2 : both childs are girls : F/F
Setup 3 : the first child is a boy, and second child is a girl. M/F
Setup 4 : the first child is a girl, and the second child is a boy. F/M.
Why order matters in setup 3 and 4 ? Because M does not equal F, while M=M and F=F. We investigate not the individual itself, but the property of the
individual (in this case, the gender). Therefore, M/F is not equal to F/M, and in the real
world there must be a first and a second child.
If you are asked to write down all the possible setups of a family of two in the real world, you would write the same
table. You'd say that :
Some families have 2 boys = 1 setup
Some families have 2 girls = 1 setup
Some families have one girl and one boy = 2 possible setups (1st one is a girl OR a boy).
Your argument of F/M = M/F implies that all families have _either_ one of the 2 setups, every first child is a boy,
or a girl. But it doesn't work like this in real life. That is why order matters.
Conclusion : if we agree that there are 4 possible setups in a family of 2 childs, then we have a probability of
1/4 to guess the correct setup of the family given NO information. But if we are informed of the gender of
one of the child, then one solution of the setup is removed (F/F or M/M), and we have a chance of 2/3 to guess right IF we chose the opposite gender (see Monty Hall problem).
And if we are informed of the gender of one specific child (1st one or 2nd one), then it leaves us with only 2 solutions ! And here, the probability
becomes 1/2.
I'd like you to notice that your new table of probabilities imply that I have a chance of 1/6 to guess the gender setup of a family of 2 children. I don't think that makes sense either to you.
Let's make the experiment a bit clearer :
- We gather a number of families who have 2 childs.
- For each family, we announce the gender of one of the child, but we don't know wich one.
- We are then asked to guess the sex of the other child.
At this point, you believe that the probability to guess right is 1/2, and that the 2/3 probability doesn't make any sense. My claim is that you fall in the Monty Hall problem trap, wich is very counter-intuitive and doesn't seem to make sense at first.
But here is some clarification of the problem :
- What we are really asked is to guess the _gender_ setup of the family. So we need to establish the universe of possible family setups before answering. What are they ?
Even if we don't care about the order, we must acknowledge that there are 2 childs in the family, so there must be a first child, and a second child.
Setup 1 : both childs are boys : M/M
Setup 2 : both childs are girls : F/F
Setup 3 : the first child is a boy, and second child is a girl. M/F
Setup 4 : the first child is a girl, and the second child is a boy. F/M.
Why order matters in setup 3 and 4 ? Because M does not equal F, while M=M and F=F. We investigate not the individual itself, but the property of the individual (in this case, the gender). Therefore, M/F is not equal to F/M, and in the real world there must be a first and a second child.
If you are asked to write down all the possible setups of a family of two in the real world, you would write the same table. You'd say that :
Some families have 2 boys = 1 setup
Some families have 2 girls = 1 setup
Some families have one girl and one boy = 2 possible setups (1st one is a girl OR a boy).
Your argument of F/M = M/F implies that all families have _either_ one of the 2 setups, every first child is a boy, or a girl. But it doesn't work like this in real life. That is why order matters.
Conclusion : if we agree that there are 4 possible setups in a family of 2 childs, then we have a probability of 1/4 to guess the correct setup of the family given NO information. But if we are informed of the gender of one of the child, then one solution of the setup is removed (F/F or M/M), and we have a chance of 2/3 to guess right IF we chose the opposite gender (see Monty Hall problem).
And if we are informed of the gender of one specific child (1st one or 2nd one), then it leaves us with only 2 solutions ! And here, the probability becomes 1/2.