Sons and Tuesdays

I recently discussed the following problem:

You run into an old friend. He has two children, but you do not know their genders. He says, “I have a son born on Tuesday.” What is the probability that his second child is also a son?

I had heard this problem at the Gathering for Gardner 9 in a private conversation. My adversary had been convinced that the answer to the problem is 13/27. I came back to Boston from the gathering and wrote my aforementioned essay in which I disagreed with his conclusion.

I will tell you my little secret: when I started writing I substituted Wednesday for Tuesday. Then I checked my sons’ birthdays and they were born on Saturday and Tuesday. So I changed my essay back to Tuesday.

After I published it people sent me several links to other articles discussing the same problem, such as those of Keith Devlin and Alex Bellos, both of whom think the answer is 13/27. So I invented a fictional opponent — Jack, and here is my imaginary conversation with him.

Jack: The probability that a father with two sons has a son born on Tuesday is 13/49. The probability that a father with a son and a daughter has a son born on Tuesday is 1/7. A dad with a son and a daughter is encountered twice as often as a dad with just two sons. Hence, we compare 13/49 with 14/49, and the probability of the father having a second son is 13/27.

Me: What if the problem is about Wednesday?

Jack: It doesn’t matter. The particular day in question was random. The answer should be the same: 13/27.

Me: Suppose the father says, “I have a son born on *day.” He mumbles the day, so you do not hear it exactly.

Jack: Well, as the answer is the same for any day, it shouldn’t matter. The probability that his second child will also be a son is still 13/27.

Me: Suppose he says, “I have a son born …”. So he might have continued and mentioned the day, he might not have. What is the probability?

Jack: We already decided that it doesn’t depend on the day, so it shouldn’t matter. The probability is still 13/27.

Me: Suppose he says, “I have a son and I do not remember when he was born.” Isn’t that the same as just saying, “I have a son.” And by your arguments the probability that his second child is also a son is 13/27.

Jack: Hmm.

Me: Do you remember your calculation? If we denote the number of days in a week as d, then the probability of him having a second son is (2d−1)/(4d−1). My point is that this probability depends on the number of days of the week. So, if tomorrow we change a week length to another number his probability of having a son changes. Right?

At this point my imaginary conversation stops and I do not know whether I have convinced Jack or not.

Now let me give you another probability problem, where the answer is 13/27:

You pick a random father of two children and ask him, “Yes or no, do you have a son born on Tuesday?” Let’s make a leap and assume that all fathers know the day of the births of their children and that they answer truthfully. If the answer is yes, what is the probability of the father having two sons?

Jack’s argument works perfectly in this case.

My homework for the readers is: Explain the difference between these two problems. Why is the second problem well-defined, while the first one is not?

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24 Comments

  1. Tanya Khovanova:

    Here is a letter I received from Jason Rosenhouse (He allowed me to mention his name):

    Concerning your dialogue with “Jack,” I would approach the problem just by enumerating the sample space. If all I know is that the man has two children, then there are 196 possible ordered pairs representing the sexes and birth days of the children. (The first child could be either of two sexes born on any of seven days, for a total of fourteen possibilities. Likewise for the second child, which gives 196 pairs.).

    The effect of learning that one child is a son born on Tuesday is to dramatically reduce the sample space. There are only 27 ordered pairs in which one of the children is a son born on a Tuesday. (First child is the son born on Tuesday paired with fourteen possibilities for the second child, or the second child is the son born on Tuesday paired with fourteen possibilities for the first child, minus the one case we double counted.) In 13 of those 27 cases the second child is also a son. Hence the 13/27 answer. I think it is pretty clear what assumptions I am making to justify that answer.

    If you had told me instead that one child was a son born on Wednesday then I would do the same calculation. But if the person told me had a son born on ????, and simply mumbled the day, then I would no longer be able to narrow down the sample space. To put it another way, the father’s mumble did not give me any information I did not already have. I would learn something from knowing that one of the children was a son but the mumble would contribute nothing. So I would not answer 13/27 in this case.

    Instead I would say that of those 196 pairs I mentioned earlier, there are 49 where both children are girls. That leaves 147 pairs where at least one of the children is a son. The second child is a son in 49 of those pairs. So the probability would be 49/147 = 1/3, which is the answer to the traditional form of this problem.

  2. Tanya Khovanova:

    Jason,

    There are 196 possibilities. To calculate the distribution of the cases when the father can say “I have a son born on Tuesday,” we need to multiply each case on the conditional probability that given this case the father says the sentence above.

    If the father has two daughters the probability that his says, “I have a son born on Tuesday,” is zero. If the father has a son born on Tuesday and a daughter born on Wednesday what is the probability that he says “I have a son born on Tuesday”? The situation is uncertain. We do not know this probability. In your calculation you implicitly assume that this probability is 1. There is no any reason for these assumption. Not to mention that it is biased towards sons and thus politically incorrect.

  3. misha:

    Tanya, we don’t have to consider all the 196 possibilities in the first problem because our friend has already said that he had a son was born on Tuesday. If we assume that this information is correct, Jason’s calculation holds, and if we doubt this information, all the bets are off.

  4. Tanya Khovanova:

    MIsha,

    Jason is missing a step. Given that the father has a son born on Tuesday, what is the conditional probability that he says that he has a son born on Tuesday?

  5. misha:

    But we already know that he said that, so the conditional probability you mentioned is out of the picture. It is a nice problem, but I would like even better the problem of roughly estimating the probability without any computations. It’s clear that the answer will be close, but a bit smaller than 1/2, because twice as many people with 2 children and 1 son have only one son, but people with 2 sons have almost (but not quite, because of the repetition of the days) the chance to have a son born on Tuesday, therefore…

  6. Tanya Khovanova:

    Misha,

    We know that he said that, but we do not know why. This creates an uncertainty. Thus we need a conditional probability that he said that given that his son was born on Tuesday. Are you saying that under no circumstances the person who has a daughter born on Wednesday would never say so?

  7. misha:

    Do we have to care why he said he had a son born on Tuesday? It is given in the problem and assumed to be true. So we can safely start with all the people who have 2 children one of which is a boy born on Tuesday. Among these people 14/27 have a boy and a girl and 13/27 have 2 boys.

  8. misha:

    When I said “… one of which…” I really meant “… at least on of which…,” not “… exactly on of which…” I hope it’s clear.

  9. misha:

    As I see it, why he said it is irrelevant, as long as what he said was true.

  10. Tanya Khovanova:

    I just posted a superb essay by Peter Winkler on the subject:
    https://blog.tanyakhovanova.com/?p=234

    Misha, it might answer your questions.

  11. misha:

    I haven’t read the essay yet, but I have a hunch of what you were driving at. I guess your point was that the very fact that the man had blurted out that he had a son born on Tuesday could be an indication that he had some preference for the sex of his children, and therefore he was not an average Joe that would fairly represent the herd. Looks a bit far-fetched to me. It is true that there are ways to control the sex of your children, short of abortion or infanticide, that are fairly reliable. But it is not clear what preference was indicated by this character volunteering the information. By the way, picking a random father of 2 children in your second problem runs into a similar difficulty. Conducting a survey is a better, but even then you would have to be careful about your sampling.

  12. Bill:

    It seems to come down to who decided there was something special about Tuesday, the interviewer or the father.

    Case 1: The interviewer is the first person to mention Tuesday. She asks the father if he has a son born on Tuesday, and the father says yes. 7 of 49 fathers with a boy and a girl will say yes, 7 of 49 fathers with a girl and boy will say yes, and 13 of 49 fathers with two boys will say yes. Therefore, 13 out of every 27 fathers who say yes will have two boys.

    Case 2: The father is the first person to mention Tuesday. The interviewer asks the father to identify the day of birth for his son, and in the case of two sons, to flip a coin to decide. Again, 7 of 49 fathers with a boy and a girl will say Tuesday. Again, 7 of 49 fathers with a girl and a boy will say Tuesday. And 13 of 49 fathers with two boys will have THE POSSIBILITY of saying Tuesday. Of these 13 fathers, 1 of them will have no other option and must say Tuesday (both boys being born on that day). Of the other 12, the coin flip will determine whether he says Tuesday or not. So, 6 of these fathers will say Tuesday and 6 will say some other day. So of all fathers who say Tuesday from 147 asked, 14 of them have one child of each gender as before, but this time, only 7 of them have two boys. So the chances the other son is a boy is 1/3, as expected.

    And that’s a fact, Jack!

  13. misha:

    Oh, I see now. Of those who have second son born not on Tuesday, only half would say they had a son born on Tuesday. So after all, those with 2 sons are not as heavily represented in the sample as I assumed.

  14. misha:

    Still, case 2 suggested by Bill is not quite the same as the original problem, since the coin flip is required only from the fathers of 2 boys. If we assume in the original problem that everybody flips the coin to decide the day of which child to talk about, then the probability in question will be 1/2.

  15. Bill:

    Oh, that’s a good point, Misha. In the version presented at the top of the post, the father is the first person to mention Tuesday, but he is also the first person to mention gender.

    So 0 of 98 fathers with two daughters will say “I have a son born on Tuesday.” 14 of 98 fathers with a son and a daughter will have a son born on Tuesday but only 7 of them will say it, and by the same logic 7 of 98 fathers with a daughter and a son will make the statement. Of 98 fathers with two sons, 26 will have at least one son born on a Tuesday. 2 of those must say “I have a son born on a Tuesday” as that describes both children. Of the remaining 24, 12 of them will talk about the son born on Tuesday while the other 12 will talk about the son born on a different day.

    Therefore, of 392 fathers of two children, 28 will make the statement “I have a son born on Tuesday.” Of these, 14 have a son and a daughter, and the other 14 have two sons.

    So with no information about why the father made this unsolicited statement, the answer must be 1/2, as Misha points out.

  16. misha:

    Actually, the answer 1/2 becomes obvious when each child has an equal chance to be mentioned.

  17. Peter Lawton:

    The “Gathering for Gardner” problem is a hoax. The mention of Tuesday is there to lead you into a trap.

    The 13/27 answer, though perfectly correct, is the answer to a different problem from the one stated.

    It is the probability within the “Boy born on Tuesday” subset.

    We are asked for the probability within the whole set (2 children, one of which is a boy), not the probability within the subset “2 children, one of which is a boy born on a specific day of the week”

  18. Peter Lawton:

    …correct answer to the question stated is 1/3

  19. Peter Lawton:

    Bill (see above) has got it.
    The one-boy fathers are very over represented in the group who volunteer “Tuesday” for the day a son was born. Half of the two-boy fathers with a son born on Tuesday will volunteer the birthday of the other son.
    13/27 doesn’t apply, therefore.

  20. Francis Davey:

    Tanya,

    Your analysis on https://blog.tanyakhovanova.com/?p=221 was really great. It articulated much better than I was able to the problem with the standard 13/27 answer and why it might seem unintuitive (because it is sometimes not the right answer). I had a look at other blogs/discussions and yours was the first that properly made sense. Much obliged.

    Francis

  21. Peter Lawton:

    Yes, your analysis makes it all clear.
    One thought – if you assume each of the three ways in which the information could have been obtained has the same probability (the default in the light of no information?), shouldn’t you end up with a final, overall probability for the man being a BB?

  22. Tanya Khovanova:

    Peter,

    If I have a probability distribution in my head of what a person might say in different situations, I can calculate the final answer. If my friend says “I have a son born on a Tuesday” and nothing else, I might think that the other child is very probably a daughter, because otherwise he would have probably said: “One of my sons is born on a Tuesday”.

  23. Eliminating the irrelevant:

    […] her Math Blog, Tanya Khovanova approaches the problem in a very interesting way. Paraphrasing Tanya, if Mr Smith […]

  24. Alkis Piskas:

    Actually, I don’t get it … What does the day of the week has to do with everything?
    Suppose the friend just says “One of them is a boy.” and I ask, “What day of the week is he born?” and I get a day, do I have more data for the problem??? The boy is certainly born a day of the week! (Except if he was born on Joshua’s missing day!)
    So, I don’t see how this is different from the classic probability problem, to which the answer is 1/3 (i.e., one of 3 possible cases boy-girl, girl-boy, boy-boy).