Probability that new baby born is a boy in a nursery There are 2 boys and unknown number of girls in a nursery. A new baby is just born inside the room. We pick randomly a baby from the room, it turns out that the baby is a boy. What is the probability that the new baby just born is a boy?
We can solve this using Bayes' rule, as follows, P(new baby is boy | picked a boy). We end up getting 3/5. Why is this answer intuitively not dependent on the number of girls in the nursery?
 A: Suppose there are initially $g$ girls in the room.  Let $A$ be the event that the new baby is a boy, $B$ the event that the baby picked is a boy.  I presume we are supposed to assume $\mathbb P(A) = 1/2$ (although in real life that is not quite true).  Then $\mathbb P(B\mid A) = 3/(g+3)$ and $\mathbb P(B\mid A^c) = 2/(g+3)$,
so $$\mathbb P(A\mid B) = \frac{\mathbb P(B\mid A) \mathbb P(A)}{\mathbb P(B\mid A) \mathbb P(A) + \mathbb P(B\mid A^c) \mathbb P(A^c)} = \frac{3}{5}$$
The point is that $g$ has the same effect on both $\mathbb P(B\mid A)$ and $\mathbb P(B\mid A^c)$, a multiplicative factor of  $1/(g+3)$ in both cases, which cancels out in $\mathbb P(A\mid B)$.
A: Before the new baby is born, there are 2 boys (Andy and Bob).  The new baby (NB) is born, then we pick a baby at random and get a boy.  In order for this to happen, one of the following scenarios occurred:

*

*NB is a boy and we picked Andy


*NB is a boy and we picked Bob


*NB is a boy and we picked NB


*NB is a girl and we picked Andy


*NB is a girl and we picked Bob
In three of these five cases, NB is a boy, so the probability is $\frac{3}{5}$
The number of girls at the start is irrelevant, because changing that number doesn't change the relative probability of any of scenarios $1-5$ occurring
