A Son that is Born on Tuesday

You meet a guy talking about his two kids, and he mentions a son.

 What is the probability that he has two sons?

This is a classic math problem; to spoil the solution, the probability that he has two sons is only 1/3. The list of possible options that could be the arrangement of his children is (Son, Son), (Son, Daughter), and  (Daughter, Son). Of these, each is equally likely. 

Here I would like to discuss a variant of this problem:

You meet a guy talking about his two kids, and he mentions a son. You find out that that child was born on a Tuesday.

What is the probability that he has two sons?

Any reasonable person should read this and the above and say 1/3, because of everything we said above. But some Professors have been perplexing the populace with their priors. 

They claim that the answer is 13/27ths, using the same logic as above, e.g., for any gender child + day of the week as x, our options are now: (Son+tuesday, x), (x, Son+tuesday). Don't forget to count (Son+tuesday, Son+tuesday) only once here. Of these 27 realities, only 13 have the two sons involved. 

Of course, this logic is incorrect. The reason is that there is no hypothesis for the day of the week that someone is born on. A hypothesis is an initial guess made before the outcome of an experiment. In this case, we must assume (as, to my knowledge at least, no part in white American culture makes being born on a Tuesday any more notable than being born with an odd or even number of hair on the head) that no such hypothesis has been pre-given.  To discuss this further, let's ask a different question:

I just rolled (at 7:53pm 9/2/2026) a twenty 6-sided dice(virtually). One of the following lists, A or B, is my result. Which do you think it is?

A. 6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6

B.4,4,5,3,3,2,1,3,3,6,4,6,5,1,6,2,4,2,6,6

Note that both outcomes are equally likely. From a purely mathematical perspective, as these are different dice, there is nothing that would separate outcome A from outcome B. Explicitly, one would have to associate some value to 6 on one die and a 6 on the other, but there is no such association being given from a pure mathematical perspective. Yet I think that any given reader can easily find out which outcome is true. The reason this is the case is that there is a silent hypothesis that exists among all people who have interacted with dice. Namely, that rolling a bunch of the same number in a row is a notable event, the highest possible value even more so. We have naturally added a structure to these dice and mapped something between them. Similarly, if I told you that I just did something that had an infinitesimal chance of happening and described to you outcome B, there is a feeling that you have been lied to. Specifically, because while an event with an infinitesimal chance did happen, the lack of hypothesis means that I did not do said event. 

The event is not notable unless a hypothesis was chosen prior to the outcome.

So now let's go back to the Tuesday Son and ask ourselves: are we allowed to do statistical inference on the gender of the children using the information that the birthdate is a Tuesday? Specifically, we know that the birthdate was one of the days of the week, and that each of these is equally likely, and thus we could equally have this information be true for any day, Sunday through Saturday. We chose Tuesday without loss of generality in making the problem. Say a father was proud of the idea of having two boys, yet instead has a son and a daughter. During the son's birth, he makes note of the hour, minute, and second of birth. Now, whenever mentioning that he has a son, he mentions this degree of information. Should you be tricked into believing that he has a higher than 33% chance of having two sons? My conclusion is that a proper understanding of the philosophy behind Bayesian probability implies that, without a real hypothesis making having been born on a Tuesday special, this cannot change the probability of the event specifically because it is not a hypothesis, and the probability stays at 1/3 because gender of a baby is an obvious hypothesis.

When does this 13/27 trick actually hold?

The fully correct answer to this is that if we were to select families based on the fact that they had a son that was born on a Tuesday and counted the number of times that they had two sons the answer would be 13/27. But if you talk to a guy and he tells you about his son and a bunch of cool things about the son that he thinks are neat (but are in fact quite mundane and random) the probability has not changed for the chance that he has two sons (unless that was one of his fun facts). However we do have the following case:

If the guy later tells you that he was born on a Tuesday and his father was born on a tuesday and so on for 50 generations, then you can confirm that the probability that he has two sons is again 13/27.

Simply, he has to be talking about his son because there was at least one son born on a Tuesday, for us to change how we evaluate any probabilities. 

Or more abstractly, evidence is about how likely your observation is under different hypotheses. These hypotheses have to be defined before the event for a valid study. Evidence is not just how rare the observation is; you must care about the observation beforehand if you want to use it to update your probabilities.