Assume the probability of having a boy and a girl are equal (50%).
From a given population, pick at random a family that has exactly 2 children, one of whom is a boy. What is the probability that the other child is a girl?
or word trickery?
The correct answer is 66.7%
Families with exactly 2 children are going to be equally distributed among the following groups
Group A: Girl, Girl
Group B: Girl, Boy
Group C: Boy, Girl
Group D: Boy, Boy
Since group A is excluded, we randomly pick a family
from groups B, C and D. Therefore the probability of picking a family with a girl and a boy is twice as big as the probability of picking a family with 2 boys.
Notice how the question says the probability of "the other child" being a girl. In case of B and C, we only have one option of picking "the other child". For D, either of the children could be the "other".
This question shows the importance of correctly picking a sample from a population when doing statistical analysis. If somebody were to pick a sample of families
as described above, he might reach the incorrect conclusion that girls are more common than boys.
Consider this question instead:
"From a given population, pick at random a boy from family that has exactly 2 children. What is the probability that the other child is a girl?"
In this case, the answer is 50%
We're picking a boy at random, and not a family, so for a family of two boys, each boy has a chance of being picked. Meaning that picking a boy from family D will be twice as likely as that of family B or C.