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You work for the census bureau. You call a random phone number and learn that the household has exactly two children and at least one of them is a girl. What is the probability that they are both girls?

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What are the different possibilities of combinations of the genders of the children?
How many are there?

Did you expect the answer to be 1/2?
Well, most people do. However, if B = boy and G = girl, then BG and GB are 2 separate cases.
They may in practical life be the same, but in context of the problem, their individual probabilities matter.
The first thing we know is that there are 2 children. Our combination (all equally likely) are:

The next piece of information we have is that at least one is a girl. Obviously, the remaining possibilities are BG, GB and GG.
Out of these, the chance that both are girls is 1/3 or 0.33.
A common question in this problem is "Why do we take BG and GB as separate possibilities?"
The answer to this is that while they may in some cases be effectively the same, they need to be taken into consideration separately when their probability is being calculated.

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