It’s so counter-intuitive my head is going to explode.
Here’s the paradox for the uninitiated:If I say, “I have 2 kids, at least one of which is a girl.” What is the probability that my other kid is a girl? The answer is 33.33%.
>*Intuitively, most of us would think the answer is 50%. But it isn’t. I implore you to read more about the problem.*
Then, if I say, “I have 2 kids, at least one of which is a girl, whose name is Julie.” What is the probability that my other kid is a girl? The answer is 50%.
>*The bewildering thing is the elephant in the room. Obviously. How does giving her a name change the probability?*
>
>*Apparently, if I said, “I have 2 kids, at least one of which is a girl, whose name is …” The probability that the other kid is a girl* ***IS STILL 33.33%.*** *Until the name is uttered, the probability remains 33.33%. Mind-boggling.*
And now, if I say, “I have 2 kids, at least one of which is a girl, who was born on Tuesday.” What is the probability that my other kid is a girl? The answer is 13/27.
>*I give up.*
Can someone explain this brain-melting paradox to me, please?
In: 4
You have to make a subtle hidden assumption to get the answer of 1/3, which is partly why the situation appears paradoxical.
Suppose you kept approaching random (honest) people and asked them “do you have exactly two children?” If they answer “no” you let them go. If they answer “yes” you ask them “is at least one a girl?”. Now if they answer “no” to that question, you know they have 2 boys, which a priori had probability 1/4. If they answer “yes”, you know they either have two girls, which has a priori probability of 1/4, or one of each, which has a priori probability of 1/2. The ratios of these scenarios must stay the same (1:2), so the probability that they have 2 girls is indeed 1/3.
Now consider this slightly different situation: your first question is the same as above. But your second question to those who have 2 children is “complete this sentence with either boy or girl so as to make it true:’at least one of my children is a …’”. Now you have two groups: all the people who complete the sentence with ‘girl’, and all those who said ‘boy’. And assuming no bias in how people answer, those groups should be the same size.
Now the first group – those who said – girl, are all people who have two children at least one of which is a girl. But the probability that the other is a girl is 50%. Because now, half the people who have both will have said ‘girl’, but the other half will have said ‘boy’.
So in the original problem, to get 1/3, we need to make an assumption as to why they said ‘girl’ rather than ‘boy’. I.e. we need to assume that they will always tell us about the girl if they have one of each. And this is, when you think about it, rather an odd assumption to make.
This is related to the Monty Hall problem, and also to the question of restricted choice in games like bridge. Information can not be considered in isolation; you also need to consider the source of the information I.e. why you received that particular bit of information rather than another. And when that isn’t random, intuitive probabilities will tend to be wrong. Eg in the Monty Hall problem, he doesn’t open a random door, he opens one he knows doesn’t contain the prize. If he opened a random one and it turned out not to contain a prize then the intuitive answer that it is 50/50 whether to swap or not would be correct.
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