The genders of the neighboring family’s children
The house next door has some new neighbors. They have two children, but you don’t know what mix of boys and girls they are. One day, your wife tells you “At least one of the children is a girl”. What is the probability that both are girls?
Your wife then tells you “The way I found out that at least one of the children is a girl is that I saw one of the children playing outside, and it was a girl”. Now, what is the probability that both are girls?
There are four possibilities for your neighbor’s children: BB, BG, GB, and GG. All of these are equally likely.
Once your wife tells you she knows one of the children is a girl, you have eliminated the possibility that the children are BB. There are three possibilities left, and the remaining possibilities are still all still equally likely, so there is a 1/3 probability that both children are children.
However, when your wife says “The way I found out that at least one of the children is a girl is that I saw one of the children playing outside, and it was a girl”, the probability changes. Using Bayes’ rule,
P(GG | child wife saw was G)
= {P(GG)P(child wife saw was G|GG)} / {P(child wife saw was G)}
= {P(GG)P(child wife saw was G|GG)} / {(P(BB)P(child wife saw was G | BB)) + (P(BG)P(child wife saw was G | BG)) + (P(GB)P(child wife saw was G | GB)) + (P(GG)P(child wife saw was G | GG))}
= {(1/4)(1)} / {(0(1/4)) + ((1/2)(1/4)) + ((1/2)(1/4)) + (1(1/4))}
={1/4} / {1/2} = (1/4) * (2/1) = 2/4 = 1/2.
So after the second bit of information, the probability the second child is a girl is 1/2, which matches our intuition
Your Answer
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