TITLE: Probability that new baby born is a boy in a nursery
QUESTION [8 upvotes]: 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?

REPLY [12 votes]: 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