A person has two children. You do not know they are sons or daughters. Consider the folliwng two scenario
you auires some information

Scenario 1: One day he brings a boy to visit you and
introduce the boy to you as his son

Scenario 2: One day you make a phone call to him and
he said only he and his two children are at home. You
hear some boy's voice among backgroudn noise.

Now the question is to estimate teh probability
of the guy having two sons:

(1) Scanario one: You see a boy as his son. The chanve
of the other kids being a boy is 50%. The probablity
the guy has two sons is 0.5.

(2) Scenario 2: You heard a boy's voice so yiou know
at least one of his children is a boy so use the defnition of coonditional probability

P(Both are boys|at least oen boy)

=P(Both are boys and at least one boy)/P(at least one boy)

=P(Both are boys)/P(at least one boy)

=P(Both are boys)/(P(two boys or Girl and Boy in that order or Boy and Girl in that order))

=0.5*0.5/(0.5*0.5+0.5*0.5+0.5*0.5)
=1/3.

So the difference can only be caused by
different amoutn of information
given by two scenarios. But how?