Understand Bayesian arguments!

I just stumbled over another problem of probability theory on the channel MindYourDecisions. The video is rather old already, but mathematical problems never age.

„A nursery has 3 boys and a number of girls at one day, plus an additional birth at night. The next day a child from the day before is picked at random, and it is a boy. What is the probability that the child born at night is a boy?“

I hate the form that this question for a probability is posed. For the casual audience this does not make much sense. They will just answer 50%, and I would not be angry about this answer. It is sort of correct. You need a mathematical education to understand the problem to start with. Let me explain.

Everyone who has not met this kind of problems before will argue that the probability of the night child being a boy is 50%. That is true. The fact that the survey picks a boy the next day does not change this at all. I mean, the randomly picked boy was a boy. So what?

The problem is that the question is asking something more involved. It asks you to consider all possible random experiments. I.e., you assume a random child at night and a random pick. Then you discard those experiments that do not fit to the story.

An extreme case would be that there are no boys at daytime. Then a girl at night would not fit to the story, and all experiments that fit feature a boy at night. The Bayesian mathematics calls this a 100% probability for a boy.

For the actual case of three boys at daytime, let us assume g girls. I now follow the explanation of the video, but with different wordings and a completely different concept of thinking.

There are two cases now.

Case 1: We have a boy at night. We expect this case to happen in half of the experiments. The chance to pick a boy the next day is then 4/(4+g). We throw away all experiments where a girl is picked, i.e., g/(4+g) of them on average.

Case 2: We have a girl at night. Again this happens in half of the experiments on average. The chance to pick a boy next day is 3/(4+g). We throw away all experiments where a girl is picked.

Now imagine both cases together. In the experiments that we did not throw away we have an overhead of 4 to 3 for case 1. Thus, the Bayesian mathematics says that the chance for case one is 4/7.

There are other ways of writing this up. But they are only shortcuts of the above thought experiment. And for me, defining a precise experiment is the clearest way to think about such problems.

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