Conditional probability with Bayes' Rule

Question

On a practice exam from statistics I encountered a very difficult exercise I couldn't manage to solve:

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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

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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..

Answer 1

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.

Answer 2

The answer is simply 1/2. If you are required to use conditional probability, it goes like this: $$P(G_2 | B_1) = \frac{P(G_2 \cap B_1)}{P(B_1)} = \frac{P(G_2) P(B_1)}{P(B_1)} = \frac{(1/2)(1/2)}{1/2} = 1/2$$

Don't confuse this question with a very similar one where all we know at the start is that at least one child is a boy. If you have seen one child, and that one is a boy, it has no effect on the probability that the remaining child is a girl; they are independent events.

[edit] Despite being down-voted on this one, I am sticking to my guns. Here is another way to look at it. Initially, the possibilities are {BB, BG, GB, GG}, where the first letter denotes the first child seen and all possibilities are equally likely, by assumption. After the first child is seen the remaining possibilities are {BB, BG}, each of which is equally likely. The event that the second child is a girl corresponds to BG, with probability 1/2.

I repeat, don't confuse this with the problem where all we know is that at least one child is a boy. In that case, after elimination the possibilities are {BB, BG, GB]; but that's different.[/edit]

Answer 3

The answer that you accepted is wrong.

Although it seems that this question asks "what is the probability that the second child is a girl given that there is at least one boy in the family?", it actually asks "what is the probability that the other child is a girl given that you randomly see one of the child and learn that it is a boy?"

Sort of wordy, but here's how to work the second (actual) case:

Let $G$ = family has one girl and one boy $,\quad$ $B$ = you see a boy child coming out of the tent

Now, $$P(G|B) = \frac{P(G.B)}{P(B)}\\ =\frac{{1/2}.{1/2}}{1/2} \\ =\frac{1}{2} $$

The underlying concept here is that the fact that you saw one of the children coming out of the tent and found out its gender is different from that of knowing the gender of one of the children beforehand. The gender of the other hidden child is independent of the gender of the random child coming out of the tent.

Mathematically,

Let $G_1$ = one of the child is a girl,

$B_2$ = one of the child is a boy,

$R$ = a random child that you see coming from the tent is a boy

So, $$ P(G_1) = 1/2, \quad P(B_2) = 1/2, \quad P(R) = 1/2 \\ P(G_1 \cap B_2) = \{Girl Boy, Boy Girl\} = \frac{2}{4} \\ P(G_1 \cap B_2 | R) = \frac{P(G_1 \cap B_2 \cap R)}{P(R)}\\ =\frac{(2/4).(1/2)}{1/2} \\ = \frac{1}{2} $$

Answer 4

Let $S$ refer to the child seen. Let $BG$ refer to the event that the family has one boy and one girl (i.e. I don't mean to distinguish $BG$ and $GB$). Then a minor variant of Bayes' rule gives

$$ P(BG | S=B) = \frac{P(S=B | BG) P(BG)}{ P(S=B | BG) P(BG) + P(S=B | BB) P(BB) + P(S=B | GG) P(GG)} = \frac{P(S=B | BG) \cdot \frac{1}{2}}{P(S=B | BG) \cdot\frac{1}{2} + 1 \cdot \frac{1}{4} + 0 \cdot \frac{1}{4}} = \frac{2 P(S=B | BG)}{2 P(S=B | BG) + 1} $$

so to answer the question, you need to decide upon the probability that you would see the boy if it happened to be boy-girl.

If you decide $P(S=B|BG) = 1/2$, then $P(BG|S=B) = 1/2$.

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What you tried to compute is incorrect;

$$ P(\text{one is a girl} | \text{one is a boy}) = \frac{P(\text{one is a girl and one is a boy})}{P(\text{one is a boy})} = \frac{P(BG)}{P(BG) + P(BB)} = \frac{1/2}{1/2 + 1/4} = \frac{2}{3} $$

The problem is that this approach discards information: specifically that one child was seen, and it was that child that was a boy.

More explicitly, this calculation also includes the possibility that the family has one boy and one girl, and you saw a girl come out of the tent in the morning.