I have a question regarding how to form the sample space in this famous paradox.
Usually the sample space is defined as (B,B), (G,B), (B,G) and (G,G). However if I express it as (B1,B2), (G,B) and (G1,G2) then when I observe one of the kids the options are:
| first observed | not observed |
|---|---|
| B1 | B2 |
| B2 | B1 |
| B | G |
| G | B |
| G1 | G2 |
| G2 | G1 |
In this case, if in my first observation a find a boy, only the following options are possible:
| first observed | not observed |
|---|---|
| B1 | B2 |
| B2 | B1 |
| B | G |
And this means that the probability of the other sibling being a boy is 2/3.
With this approach the result is quite different, so I may be solving a different problem, but I can't explain why.
Any ideas?
The problem you are solving is if that if the three situations 2 boys, 2 girls, and a boy and a girl are equally likely, and you randomly pick one of the children and observe it is a boy, then the conditional probability the other is a boy is 2/3.
But if, as is the usual assumption, the babies are generated by independent coinflips, then the situation where there is a boy and a girl is twice as likely as the other two, and then if you picked one at random and observed a boy, then the probability the other is a boy is 1/2.
This is different from if you are simply told one is a boy, in which case the conditional probability of the other being a boy is 1/2 (under your prior distribution) or 1/3 (under the usual prior distribution).