On a practice exam from statistics I encountered a very difficult exercise I couldn't manage to solve:
In the tent next to you there is a family with two children. Early in the morning you see a boy coming out of the tent. What is the probability that the other child is a girl?
Use Bayes' Rule
My approach to the solution was the following:
We assume $P(GIRL)$ = 0.5 and similarly $P(BOY)$ = 0.5.
We have to compute the following conditional probability: $P($One child is a girl| One child is a boy).
By applying Bayes' rule we should be able to compute this probability.
Bayes Rule: $P(A|B)$ $=$ $\frac{P(B|A)*P(A)}{P(B|A)*P(A) + P(B|A^c)*P(A^c)}$
Could anyone please help me with this, I tried many things but nothing worked out..
Best Answer
You can obtain two answers to this, actually. The problem is called the Sisters' Paradox. See this excellent explanation.
The most common solution, I would say, goes as follows. Let $G$ denote a girl, and $B$ a boy such that $P(BG)$ means probability of a girl and a boy. $P(GG)=P(BB)=1/4$, $P(BG)=1/2$. Conditioning on a boy($P(B)$):
$$ P(BG|B)=\frac{P(BG)}{P(B)}=\frac{P(BG)}{P(BB)+P(BG)}=\frac{1/2}{1/2+1/4}=\frac{2}{3} $$
Note that in the reference this way of solving the question yields $1/3$, but that's because in that case it's $P(GG|G)$ (or, equivalently, $P(BB|B)$) rather than one of each. But, $P(BB|B)=1-P(BG|B)=1-2/3=1/3$, so the answers are in spirit the same - just different formulations.