They are right, and you are wrong. Here is an easy way to test this.

An equivalent problem:

If someone threw a coin twice, and all you know is that at least one of those throws ended up tails, what is the probability that they got tails tails?

It is a trivial matter to write a script which performs for example a million pairs of coin throws, and then divide the number of tails tails by the number of pairs where at least one is tails. And you get 1/3.

The whole point of the problem is that order doesn't matter. If you knew the first child was a boy, then it would be fifty-fifty for the next child to be a boy. But since you don't know which one is a boy, it's 1/3 for both to be boys.

Here is the script I wrote to test this.


#!/usr/bin/env python

from random import randint

A = []

for x in xrange(100000):
A.append([randint(1,2), randint(1,2)])

c = 0
d = 0

for a in A:
if 1 in a:
c += 1
if a == [1,1]:
d +=1

print float(d)/float(c)