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In a sampling of 1000 families, the expected values of each kind of family is as follows: 2xB : 250 1xB, 1xG: 500 2xG : 250 Sampling this population ign
by tel 16y ago
In a sampling of 1000 families, the expected values of each kind of family is as follows:
2xB : 250
1xB, 1xG: 500
2xG : 250
Sampling this population ignoring any family that has no boys leads to the probabilities
2xB : 1/3
1xB, 1xG: 2/3rds
You're still looking at the same probabilities; the models agree.
I don't fully understand your two random variables formulation. I think the confusion you're getting at is that there is an assumption that the chance of any given birth being male is theta = 0.5. The question however is not
"I have two children, at least one is a boy, what is the probability that my next child is a boy?"
It instead has to do with binomial probabilities on the space of a few repeated trials under parameter theta. The distribution is no longer flat.
Here's a more stark example of a similar form.
"I have 300 children, and at least 1 is a boy. What are the odds that I have no girls?"
- iliketosleep 16y agook, you stated this very well. It's now clear where the confusions arises: "Sampling this population ignoring any family that has no boys." Yes, with this interpretation the answer is 1/3, but it's contrary to my interpretation. Actually, for anyone who is interested, see "Boy or Girl paradox". There is literature on this which discusses the different interpretations.
- tel 16y agoThat sampling arises because being part of the population which has no boys is * necessary and sufficient* to (truthfully) make the statement that forms the paradox. The alternative interpretation of the paradox arises when the wording of the paradox is construed to identify one of the children as male or female. In this case (stating something like "my first child is male"), being part of the population (x \in {BB, BG}) is necessary and sufficient and leads to the 1/2 probability of having two boys. In short, the question becomes whether you believe the child is identified in the wording of the question. Honestly, the author of the paradox goes pretty far out of their way to say "at least one of the children is male" avoiding that identification.