Bayes theorem and conditional probability example.

Question

Greetings to all.
I started an MSc that includes a course in statistics. Full disclosure: my bachelor’s had no courses of statics and it is in biology.

So, the professor was trying to explain the Bayes theorem and conditional probability through the following example.
“A friend of yours invites you over. He says he has 2 children. When you go over, a child opens the door for you and it is a boy. What is the probability that the other child is a boy as well.”

The math say the probability the other child is a boy is increased the moment we learn that one of the kids is a boy. Which i cannot wrap my head around, assuming that each birth is a separate event (the fact that a boy was born does not affect the result of the other birth), and the result of each birth can be a boy or a girl with 50/50 chance.
I get that “math says so” but… Could someone please explain? thank you

In: 77

Answer 1

Your professor is not applying the Bayes theorem correctly, and your gut feeling is right.

The Bayes theorem states that P(A|B) = P(B|A)*P(A)/P(B), where P(A) and P(B) are the probabilities of events A and B respectively and independently of each other without any other given conditions, P(A|B) is the probability of event A given that B is true, P(B|A) is the probability of event B given that A is true.

So let’s say that event A is both children being boys and event B is a boy opening the door. Then P(A) (the probability of both children being boys without any given condition) is 0.25, P(B) (the probability that a boy answers the door without any given condition) is 0.5, and P(B|A) (the probability that a boy answers the door given that both children are boys) is 1. Therefore P(A|B) – the probability of both children being boys given that a boy has opened the door – is equal to 1 * 0.25 / 0.5 = 0.5. This is the correct application of the Bayes theorem to the problem.

Edit: Also this is known as the Boy or Girl Paradox, and the answer to the problem really depends on the exact phrasing of how the information about the sex of at least one of the children is obtained: https://en.wikipedia.org/wiki/Boy_or_Girl_paradox