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

Here’s the way I see it.

For notation purposes, the child listed first is the one who opens the door, and the child listed second is the one who doesn’t. In the case where any child answers the door, the possible cases are as follows. In the cases where both children are the same gender, 1 denotes the older child, and 2 denotes the younger child.

1. B1B2

2. B2B1

3. BG

4. GB

5. G1G2

6. G2G1

Then, after knowing the given information that a boy opens the door, cases 4-6 are no longer considered. This leaves cases 1-3.

Among cases 1-3, two of them involve a boy, and only one of them involves a girl. Therefore, the chance that the parent has two boys given that a boy opens the door is **2/3**.

Since the question your professor asked is what the probability is that the unseen child is a boy *as well*, the “as well” part makes the probability that is being asked already take into account the given information. The 50% answer would only be true if the question didn’t specifically include “as well”; in that case, you’d indeed not care about the gender of the child who opened the door at all.

Let me know which part of this breaks down.