You know your friend has two children, and you know one of them is a girl. Suppose the probability of each sex of each children is simply 1/2 (i.e. ignoring biological and human/society trivial), what is the probability of both children are girls?
There is actually an explanation of this question on Wikipedia (Boy or girl paradox). The answer really depends on how you understand the words "you know one of them is a girl".
Interpretation 1:
Suppose we call the two children A and B as we don't know them, then the interpretation is "either you know A is a girl, or you know B is a girl". This is also equivalent to say that "you happened to see one of them and thus knew that it is a girl, but you don't know if it was A or B you saw".
From this interpretation, we may give the answer 1/2, i.e. the possibility of "both the two children are girls"=="the other child is also a girl" is 1/2.
Interpretation 2:
Still suppose we call the two children A and B, but this is unimportant for this interpretation. The interpretation is that we know nothing but "at least one of A and B is a girl".
From this interpretation, we may give the answer 1/3, i.e. the possibility of "both the two children are girls" is 1/3.
Using Bayes' theorem we also clearly see both interpretations make their own sense.
From Wikipedia:
Martin Gardner, the original author of this problem, later acknowledge that the second interpretation was ambiguous, because you must clarify how people get to know that "at least one of A and B is a girl", and the way people get to know this is likely to be the same way in the first interpretation.
Martin Gardner, the original author of this problem, later acknowledge that the second interpretation was ambiguous, because you must clarify how people get to know that "at least one of A and B is a girl", and the way people get to know this is likely to be the same way in the first interpretation.
However there are realistic ways to get the second interpretation. For example you may spot your friend buying girl's toys and get confirmed they are buying them for their own kids, or else, you may have heard that you friend once saying "Thank god I don't have two boys".
Then how we decide to use interpretation 1 or 2? It really depends on how the question is phrased. There are a few opinions to distinguish them if you are asked a similar question:
1. Whether it is the family (parent) or one of the children that is randomly selected and watched. The answer is 1/3 for the family and 1/2 for the children.
2. Whether or not there is "identilisation" of the two children, no matter if it is implicit or explicit, ordered or unordered. Usually if the phrase "at least one of them" appears, we know there is no identilisation and the answer should be 1/3. Otherwise the answer will be 1/2.
3. Whatever your examiner/boss/supervisor decides:P
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