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

Plenty of discussion here on the specific example, but let me back up and ELI5 Bayesian thinking.

What does it mean to say something has a certain “probability”?

In one way of thinking, we would say “probability is how often event X happens if we repeated the experiment a lot of times.” This is called frequentism. It looks at the situation independent of other information.

In another way of thinking, the probability of something is “based on everything I already know, and this new data, how likely is event X to happen?” That’s Bayesian thinking.

It’s not always intuitive. But most of us are Bayesian if we are honest—and that’s not bad. If I were to say, “How likely is it that Joanna is a firefighter?”, a frequentist might say “0.04%” by dividing the total number of firefighters by total number of adults in the US. But a Bayesian might say “0.0002%”. Why? Because the Bayesian uses their prior knowledge, which is that “Joanna” is usually a girls name and 96% of firefighters are male. In other words, the frequentist uses only the data on hand. The Bayesian combines the data on hand with prior knowledge.

Now, there are some potential problems with the professors version (or your memory of it), as others have said. But what he is trying to show is that revealing a bit of info should change the probability you expect for the coming events.