## On the “Son born on a Tuesday” probability puzzle

**If someone tells you “I have two children. One of them is a boy born on a Tuesday”, what is the probability that he has two sons?**

The only thing we need in order to solve this problem is an interpretation. Here is the most natural one.

There is a crowd of people who have exactly 2 children. We pick one person at random and ask “do you have a son born on a Tuesday?”, and then ask “do you have 2 boys?”. If the answer to the 1st question is “yes”, what is the probability that the answer to the 2nd question is “yes”? (I omit the trivial assumptions about all outcomes being equally probable).

Once we have this interpretation, we can give this problem to any kid who knows how to multiply 14 by 14.

Here is a scan of the solution that my son (born on a Monday) came up with. Please pardon his chicken scratch:

The filled squares are families that have a tuesday boy. The top left quadrant are boy-boy families. The answer is the ratio of the number of filled squares in that quadrant to the number of all filled squares. That is **13/27.**

Yes, I explained to him about conditional probabilities and gave some examples, but only to show why such problems are so unintuitive.

*"Tuesday son"** is an anagram of “so unsteady” and “yes astound”.*

*I have to confess that the purpose of this post is to test the hypothesis that blog posts that talk about this problem attract a lot of traffic. So in case it turns out to be true: dear reader, you may also find the following topics interesting:*

*https://bbzippo.wordpress.com/2010/06/02/an-unfair-game/** – guessing hidden numbers using a random generator*

*https://bbzippo.wordpress.com/2010/05/19/optional-stopping/** – a theorem about timing the market*

*https://bbzippo.wordpress.com/2009/11/05/expected-number-of-coin-tosses/** – What is the average number of coin tosses needed to throw a head?*

*https://bbzippo.wordpress.com/2009/11/25/expected-number-of-tosses-general/** – What is the average number of coin tosses needed to throw n heads in a row?*

*https://bbzippo.wordpress.com/2009/11/05/probability-of-bias/** – making a conclusion about the coin being biased based on a single throw.*

*And don’t forget to check out Xworder 🙂*

I think interpretation lets you *define* the problem before there is even a talk of solving it. Without proper interpretation we’ll have the following “paradox”:

If someone tells you “I have two children. One of them is a boy”, what is the probability that he has two sons? Clearly it’s 1/3. But if the person tells you “I have two children. One of them is a boy born on a Tuesday” the probability is 13/27. You can even pose the question in terms of how much should you bet on the person’s having two sons in case of either declaration.

Of course proper interpretation resolves this: You need to define what probability (space) is in each case. In “Tuesday” case if you know that every time you are asked to bet the person will say “One of them is a boy born on a Tuesday” then you should bet 13/27 or less. If the person will merely state the day of the week his son was born then you bet 1/3 or less. No paradox.

These sort of things are discussed in Taleb’s books Fooled By Randomness and Black Swan: i.e. without multiple trials is there even a meaning to the term probability?

NoTuesdaySon09/01/2010 at 3:43 pm

Agreed.

With regards to this particular problem, most people see the paradox in what “the sex of a child cannot depend on the day of birth”. In this respect, the two problems (with and without specifying the day of week) are not different. If you don’t specify the day, they still may say “the sex of one child cannot depend on the sex of the other one”.

Thing is, the condition gives you some information about both children: it says “one of them”. It’s just a semantic paradox: by saying “one” we give information about the two. So the new probability space defined by this condition is larger than the one we would end up with if we got the same information about just one child (e.g. the older one).

The point of my post is that we don’t really need to discuss all that in order to find the answer. We just need to count the squares.

bbzippo09/01/2010 at 4:33 pm

I disagree that the interpretation given above is “the most natural one.” But NoTuesdaySon is right, that viewing it in terms of a game can add insight. Say that on a game show, the emcee asks for volunteers who have exactly two children. The first, Ann, walks unto the stage and reads a sign that only says “Instructions for the Two Child Game” on the side facing the audience. She thinks for a moment, and says “I have a boy born on Tuesday. What is the probability that I have two boys?” Let’s assume for a moment that 13/27 is correct.

The second, Bob, repeats what Ann did except he says “I have a boy born on Friday. What is the probability that I have two boys?” 13/27 cannot also be correct here, because it was correct for Ann only if the instructions told her to mention a Tuesday Son if it was at all possible. It can be at most 11/25 for Bob, and it can only be that high if “Friday Son” was second in priority to “Tuesday Son.” Because in the two cases where Bob has both a Tuesday Son and a Friday Son, he would have mentioned the Friday Son. And it will be different for every other day of the week, which is a paradox. It can’t be different for different days.

Essentially, 13/27 is correct only if the fact “one is a boy born on Tuesday” was chosen first, and a random family was chosen from all families that fit that fact. It is far more natural to interpret this problem as randomly selecting a family first, and then randomly observing a fact that applied to them. In fact, it is because it is more natural to interpret it this way that people find it unintuitive that adding “born on a Tuesday” should change the answer.

To calculate the probability, you should not merely count the boxes in your drawing. You should write a probability in each box representing the probability a parent of that family would say “boy born on a Tuesday,” and sum the probabilities. One box, representing two Tuesday Sons, gets a 1 this way. Let’s represent all of the other shaded boxes get the variable P. The totals are (12P+1)/(26P+1). If P=1, representing your interpretation that “Tuesday Son” was required, that evaluates to 13/27. If P=1/2, representing the more natural interpretation that the two possible observations were equally possible, then it evaluates to 1/2. The “observation” interpretation is also more natural if you don’t mention Tuesday: the answer to that simpler problem is 1/(2P+1), which is 1/3 of P=1 and 1/2 if P=1/2. Note that the probability stays the same – as everybody who hears the problem expects at first – if P=1/2.

Oh, and the reason this problem generates so much internet traffic is because the people who answer 1/2 for both problems, based on intuition, do not understand enough math to express why it is more natural; and the people who understand the math usually fail to consider the ambiguity that the others sense. So neither side believes the other’s answer.

For a reference, look at http://www.mathpages.com/home/kmath036.htm. Or look up Martin Gardner’s book, where he retracted the 1/3 answer.

JeffJo10/26/2010 at 4:25 pm

Silly me, editing out a “not” in a statement without flipping the rest of it. That should be “in the two cases where Bob has both a Tuesday Son and a Friday Son, he would have mentioned the Tuesday Son.”

JeffJo10/26/2010 at 4:54 pm

Hi JeffJo,

Of course, this problem as it’s orginally stated, allows for at least 3 different interpretations. We can make assumptions about the rules of the game, about reasons why the parent mentions the particular child and the particular day, etc.

Now, why do I think that the interpretation that results in the answer 13/27 is the most natural one? I think it requires fewer additional rules and assumptions than the other interpretations. Think of this as a physical experiment: we have a detector that signals only when it encounters a parent of 2 children who has a Tuesday son. In all other cases, the detector is silent. Or if you want a “game show” interpretation: Only those participants who have a Tuesday son make it to the 2nd round, all others are sent home. No assumptions regarding human behavior.

bbzippo10/26/2010 at 6:02 pm

One could also make arguments based on information entropy (http://en.wikipedia.org/wiki/Entropy_%28information_theory%29)

It should be possible to calculate information entropy for each interpretation of the problem. I feel that will show that my interpretation uses fewer information than the others.

This should be an interesting way of thinking of conditional probability in general.

I’ll post if I manage to come up with some concrete and quantitative arguments along these lines.

bbzippo10/26/2010 at 6:24 pm

Actually, it requires more rules and assumptions. For example, your dectector has to see both children, and be pre-programmed for “Tuesday Boy.” The detector for the 1/2 answer only has to see one child, and is not pre-programmed.

But ultimately I think 1/2 is better because it uses the same solution techniques that every similar probability puzzle, except this one, uses. For example, I assume you are familiar with the “Three Prisoners” ? If you count every case where the warden could tell Tom “Dick will not be pardoned” at full value, then Tom’s chances of getting the pardon do change from 1/3 to 1/2. It is only when you allow the warden to choose 50/50 between saying “Dick will not be pardoned” and “Harry will not be pardoned,” that Tom’s conditioanl probability stays at 1/3, which is the accepted answer.

JeffJo10/26/2010 at 8:16 pm

Correct, my detector is pre-programmed for “tuesday boy”, and that is the only answer mentioned in the problem’s original formulation. In fact, I interprete “someone tell you X” as “someone gives you an affirmative answer to X, where X is a yes/no question”.

Had the problem stated anything like “the subject is talking about his/her child and makes a statement about the child’s sex and day of birth…”, I would have chosen the 1/2 interpretation.

Alright, maybe I should reformulate my claim.

The 13/27 interpretation is not the most “natural”.

But I think it is the most formal and it is the “minimal consistent interpretation”.

bbzippo10/26/2010 at 8:56 pm

Actually, “someone tells you X” and “someone gives you an affirmative answer to X, where X is a yes/no question” are quite different. The former was provided voluntarily, so X represents just a necessary condition defining the group “someone” came from. The latter was required, so X is both necessary and sufficient. It’s amazing how subtly the logical fallacy of “affirming the consequent” can creep in, but that is literally what you just said you are doing.

Or, consider two pairs of statements: “He has one boy/girl” and “He tells you he has one boy/girl.” In the first pair, each statement represents 75% of the population of two-child familes; the two cases overlap. In the second, the two statements are mutually exclusive, and can’t overlap. Since they can’t be unequal, each must be 50%.

JeffJo10/27/2010 at 10:59 am

I don’t think “affirming the consequent” has anything to do with this. I didn’t affirm anything: the problem statement told me I got this information. This is a straightforward application of the conditional probability formula. The condition is an event that has occured, is that what you are calling “affirmed consequent”?

> …“He tells you he has one boy/girl.” … the two statements are mutually exclusive…

This is English semantics. The indefinite articles “a” and “one” may or may not have the same meaning as the quantifiers “one” and “at least one”.

I believe that “He tells you he has a boy/girl” are not mutually exclusive.

bbzippo10/27/2010 at 8:43 pm

It has nothing to do with articles and/or the word “least,” which are red herrings. The point is that the event to use as a condition is the act of telling you the fact, not the fact itself. He can only tell you one thing (or else you will know the whole family makeup), so yes the two are mutually exclusive.

Let’s play a gambling game. I will roll two dice, tell you what number one die landed on, and ask you to set the odds for whether I rolled doubles of that number: is it 1 to 5 (i.e., probability 1/6) or 1 to 10 (the probability 1/11, based on 11 ways that number could exist in combination)? If we repeat this game, I’m pretty sure doubles will come up about once every six rolls.

Now, ask 100 fathers of two to tell you just one fact in the form “One of my children is a (boy/girl).” About 50 will say “boy,” and about 50 will say “girl.” 25 in each group could have said the opposite, but didn’t. 25 in each group, or 1/2 of them, will have two of whatever they said.

From the standpoint of probability, when someone randomly tells you one fact, this is the proper model to use.

JeffJo10/28/2010 at 12:53 pm

We’re back to square one.

> The point is that the event to use as a condition is the act of telling you the fact, not the fact itself

I understand that this is the key difference between the two interpretation.

> I will roll two dice, tell you what number one die landed on […] Now, ask 100 fathers of two to tell you just one fact in the form “One of my children is a (boy/girl).”

I already said that had the problem stated anything like this, I would have chosen the 1/2 interpretation. I still believe that this is not the most straightforward interpretation of the original problem.

So this whole argument boils down to

a) Semantics

b) Methodology of problem solving. What I mean is we must recognize whether this problem is (1) a probability problem that just aims to demonstrate how and why conditional probabilities may be unintuitive, or (2) an applied stats problem e.g. in the field of sociology, or (3) a game theory problem, or something else.

I recognized the problem as type (1) and chose the interpretation that in my opinion is the most suitable for this type.

I agree that other interpretations and reasoning behind them are indeed worth discussing.

That was not the goal of my original post, as I already mentioned. The goal was to present a clear solution for the particular interpretation, since I knew that many people were still confused by the unintuitive answer even after they agreed with the interpretation.

bbzippo10/28/2010 at 4:32 pm

> The point is that the event to use as a condition is the act of telling you the fact, not the fact itself

Yet another thought:

If we interprete the condition as the act of telling, we should still accept the conveyed information as an unconditioned fact.

The information is coming from the parent of the 2 children, who has full knowledge about both children, and naturally considers them both when making the statement of the form “one of them”. It would be highly unnatural to expect that the parent would conduct random sampling in order to make such statement.

On the other hand, if we receive the information from a “sensor”, it is more natural to assume that the statement could be based on knowledge about just one child.

Thank you for making me think, JeffJo 🙂

You have managed to convince me that 13/27 is indeed the most natural answer 🙂

Apparently, the formulation “the parent tells you about the children” is way less ambiguous than let’s say “we’ve learned that at least one child is a boy”.

bbzippo10/28/2010 at 9:41 pm

Well, I see you have convinced yourself, so I won’t reply after these last comments. Some of them reiterate points of mine from before, that you haven’t addressed.

1) If the intent of the original problem is to “demonstrate how and why conditional probabilities may be unintuitive,” then it needs to make the condition absolutely clear. Telling someone their answer is wrong, when it is correct based a completely valid interpretation of the ambiguous condition, defeats your purpose. And this really is the reason I try to point out the differences.

2) There are better, and more famous, demonstrations for that purpose. The Three Prisoners Problem, the Monty Hall Problem, and Bertrand’s Box Paradox are all variations of the same thing, and I choose to use the Three Prisoners as an example because it uses the same mechanism to reveal the information that Gary Foshee used. An insider to the random selection telling a fact to an outsider. But in all those problems, regardless of the mechanism, the intuitive-but-incorrect solution parallels the one you use here, and the unintuitive-but-correct one uses the methods I want to use. I think that is clear proof that the unintuitive nature of the Two Child Problem, or the Tuesday Boy Problem, is in the problem statement. Not the conditional probability.

3) Being an “unconditioned fact” does not make that fact into a sufficient condition. If it did, a parent would have to tell you every fact that was true about his family.

4) My answer of 1/2 is not based on a random sampling of the children (which would make the Monty Hall Problem a better simile), or knowledge of just one child (Bertrand’s Box Paradox), although those are ways the real basis could come about. It is based on a random sampling of the facts that apply to the family. Since the statement is coming from a parent who has full knowledge about both children, that parent had to choose between all such facts. You might be able to convince your intuition-based reader that when the difference between the facts is meaningful at some level – like between “one is a boy” and “one is a girl” – why a parent might choose one over the other. I disagree, but I can understand the argument, and I suspect it is why people seem to prefer the 1/3 answer for the two Child Problem. But not when the difference is meaningless at any reasonable level, like “born on a day of the week.” It’s a completely random fact, and must be treated as such.

5) You have ignored the fact that if 13/27 is correct, it leads to one of two paradoxes, depending on how you rationalize it. Either the probability is 13/27 regardless of what day is mentioned, leading to the conclusion that the probability has to be 13/27 when no day is mentioned; or the probability for the other six days is some ordering of 11/25, 9/23, 7/21, 5/19, 3/17, and 1/15, leading to the question “why does Tuesday get the highest value?”

6) There are two schools of thought as to what probability means. The frequentist bases it on repeatability, and will not find a difference between your problem as you stated it, and my simile of 100 fathers. In fact, that should be how he answers your question. It is the Bayesian who says probability is a measure of a state of knowledge of just one circumstance, as you do here. But a Bayesian has to re-evaluate the prior probability he assumes, based on that one circumstance, to update his probability. If you do that, using equivalent probabilities for other days of the week, the posterior probability will be 1/2.

JeffJo10/29/2010 at 12:09 pm

Thanks for the interesting discussion. You have demonstrated to me that this problem can be looked at

from many more angles than I thought.

Your comments have a lot of insight; and you are right – I have not addressed all your points. I’m planning to read through them again when I have more time, and to give them nore thought.

bbzippo10/29/2010 at 4:39 pm