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Originally Posted by TrAI
You are instructed to choose one boy from the list at random, and give the probability for the sibling being male? I would say it is 1/2, since the BB family has two boys on the list, so that family has a higher chance of being selected than one of the one boy families individually. I think this is likely to happen if you choose a child at random from a larger group of two child families too.
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That's true if you choose a
child at random, but not if you choose a
family at random.
Frog march, like the Monty Haul question, I think this one is best answered by experiencing it. Make up about 50 cards or slips of paper (you can make fewer, but as always, statistics gives the best predictions when your sample is large). For each one, flip two coins, and based on their outcome label each card. I don't care who's heads and who's tails; if you get two boys, or a boy and a girl, write it down. But if you get two girls, forget about it, and flip the coins again (we're only counting families who have at least one boy). Now, go through the cards and tally whether you have two boys or a boy and a girl, and see what result you get.
As an aside, one of the reasons this is confusing is because of the way the sample is selected. Generally, if you know of a family who has two children, at least one of whom is a boy, it might often be because you've met them. Suppose you meet a mother out walking with her son, and in the conversation, you find out that the father is somewhere else with their other child. What's the probability that the other child is a boy? It's 1 in 2, but this is
not the same selection rule as the one described, even though it may seem like it if you aren't paying close attention.