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The following probability question appeared in an earlier thread:

I have two children. One is a boy born on a Tuesday. What is the probability I have two boys?

The claim was that it is not actually a mathematical problem and it is only a language problem.


If one wanted to restate this problem formally the obvious way would be like so:

Definition: Sex is defined as an element of the set $\{\text{boy},\text{girl}\}$.

Definition: Birthday is defined as an element of the set $\{\text{Monday},\text{Tuesday},\text{Wednesday},\text{Thursday},\text{Friday},\text{Saturday},\text{Sunday}\}$

Definition: A Child is defined to be an ordered pair: (sex $\times$ birthday).

Let $(x,y)$ be a pair of children,

Define an auxiliary predicate $H(s,b) :\!\!\iff s = \text{boy} \text{ and } b = \text{Tuesday}$.

Calculate $P(x \text{ is a boy and } y \text{ is a boy}|H(x) \text{ or } H(y))$

I don't see any other sensible way to formalize this question.


To actually solve this problem now requires no thought (infact it is thinking which leads us to guess incorrect answers), we just compute

$$ \begin{align*} & P(x \text{ is a boy and } y \text{ is a boy}|H(x) \text{ or } H(y)) \\ =& \frac{P(x\text{ is a boy and }y\text{ is a boy and }(H(x)\text{ or }H(y)))} {P(H(x)\text{ or }H(y))} \\ =& \frac{P((x\text{ is a boy and }y\text{ is a boy and }H(x))\text{ or }(x\text{ is a boy and }y\text{ is a boy and }H(y)))} {P(H(x)) + P(H(y)) - P(H(x))P(H(y))} \\ =& \frac{\begin{align*} &P(x\text{ is a boy and }y\text{ is a boy and }x\text{ born on Tuesday}) \\ + &P(x\text{ is a boy and }y\text{ is a boy and }y\text{ born on Tuesday}) \\ - &P(x\text{ is a boy and }y\text{ is a boy and }x\text{ born on Tuesday and }y\text{ born on Tuesday}) \\ \end{align*}} {P(H(x)) + P(H(y)) - P(H(x))P(H(y))} \\ =& \frac{1/2 \cdot 1/2 \cdot 1/7 + 1/2 \cdot 1/2 \cdot 1/7 - 1/2 \cdot 1/2 \cdot 1/7 \cdot 1/7} {1/2 \cdot 1/7 + 1/2 \cdot 1/7 - 1/2 \cdot 1/7 \cdot 1/2 \cdot 1/7} \\ =& 13/27 \end{align*} $$


Now what I am wondering is, does this refute the claim that this puzzle is just a language problem or add to it? Was there a lot of room for misinterpreting the questions which I just missed?

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The "born on a Tuesday" question also came up here: math.stackexchange.com/questions/3278/probability-of-a-given-b/… –  Derek Jennings Sep 11 '10 at 7:20
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In case you haven't found it yet, there is a plethora of opinions here: sciencenews.org/view/generic/id/60598/title/… –  Derek Jennings Sep 11 '10 at 7:36
    
I was particularly hoping that people would directly address my derivation as written and the interpretations I used here. (Hence actually writing them out as opposed to just giving the value 13/27). –  anon Sep 11 '10 at 7:38
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9 Answers 9

up vote 15 down vote accepted

There are even trickier aspects to this question. For example, what is the strategy of the guy telling you about his family? If he always mentions a boy first and not a daughter, we get one probability; if he talks about the sex of the first born child, we get a different probability. Your calculation makes a choice in this issue - you choose the version of "if the father has a boy and a girl, he'll mention the boy".

What I'm aiming to is this: the question is not well-defined mathematically. It has several possible interpretations, and as such the "problem" here is indeed of the language; or more correctly, the fact that a simple statement in English does not convey enough information to specify the precise model for the problem.

Let's look at a simplified version without days. The probability space for the make-up of the family is {BB, GB, BG, GG} (GB means "an older girl and a small boy", etc). We want to know what is $P(BB|A)$ where A is determined by the way we interpret the statement about the boys. Now let's look at different possible interpretations.

1) If there is a boy in the family, the statement will mention him. In this case A={BB,BG,GB} and so the probability is $1/3$.

2) If there is a girl in the family, the statement will mention her. In this case, since the statement talked about a boy, there are NO girls in the family. So A={BB} and so the probability is 1.

3) The statement talks about the sex of the firstborn. In this case A={BB,BG} and so the probability is $1/2$.

The bottom line: The statement about the family looks "constant" to us, but it must be looked as a function from the random state of the family - and there are several different possible functions, from which you must choose one otherwise no probabilistic analysis of the situation will make sense.

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Nope. I'm saying that in order to give a probabilistic meaning to this statement, you need to attach probabilistic assumptions to it. In your case it seems you've chosen the assumption of "if there is a boy, a boy will be given in the statement". Other possible interpretations: "If there is a girl, a girl will be said" (so since a boy was said, there are no girls 100%), and "If there is a boy and a girl, one will be said at random" and "The sex of x is said" and so on. Forget about days - try to do the probability for these cases and see what happens. –  Gadi A Sep 11 '10 at 6:11
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This is of course correct, but you must also take into account the question of how the statement "I have a boy" comes to be. I've tried to elaborate in my answer. –  Gadi A Sep 11 '10 at 6:40
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The statement (ignoring the day red herring) is "I have two children. One is a boy." As I said before, this statement should be considered as part of the probabilistic scenario otherwise the problem is simply not well-defined. –  Gadi A Sep 11 '10 at 7:37
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What I meant was that adding the day does not affect the "strangeness" of the situation. However, I'm not sure if it's the strangeness that interests you. The main point is that it's called "a language problem" because different mathematical interpretations of the same English statement yield different mathematical models, and hence different mathematical results - and so for every intuition about the problem there is a interpretation in which the result is counter-intuitive. –  Gadi A Sep 11 '10 at 11:52
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-1 As long as we agree "One is a boy born on a Tuesday" means "At least one is a boy born on a Tuesday", then the answer is 13/27. The statement puts the family in the set { families with 2 children, at least one of whom is a boy born on Tuesday } and then 13/27 is the probability that a family picked at random from this set has 2 boys. There is no ambiguity about it, and if you're unconvinced you can run a simulation as others have done below. –  Chris Card Sep 12 '10 at 8:27
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Something still bothers me.

If someone came to me and said:

1) "I have two children. At least one is a boy born on a Tuesday. What is the probability I have two boys?"

The answer is: 13/27.

2) "I have two children. At least one is a boy. What is the probability I have two boys?"

The answer is: 1/3.

And then I see 8 fathers. The first one say "I have ... born on Sunday ..." the second father say "on Monday", etc. then for each one the probability to have 2 boys is 13/27. The eighth father say:

3) "I have two children. At least one is a boy born on some week day (but I won't tell you what day). What is the probability I have two boys?"

The answer is: ???

Probably 1/3. because we already know that the boy was born on some day. But what is the difference between him to the other 7 fathers?

Similarly, we can ask what if the father wrote the day of the birth of that boy on a card, and doesn't show it to us. How is it different from a father who show us the card?

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This is discussed also on johncarlosbaez.wordpress.com/2010/08/24/… But there is no answer there. If instead of “born on a Tuesday” your colleague says “born on—now let me think…..ah yes: it was a Tuesday.” John say: " I would argue, in the hesitation variant, that you should switch from 1/3 to 1/2 even before you hear the day: the hesitation is crucial while the day is irrelevant. For the original, unhesitant, “born on a Tuesday” case, 13/27 is correct." How the hesitation is relevant? –  Tangent Bundle May 5 '11 at 7:15
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See also this article, which also considers the same aspects of the problem. I really don't understand how someone, even after seeing the subtleties in the question, can insist that the problem is well defined and that the answer is 13/27. There is no sane reason to assume that the statement arose from a boy-centric Tuesday-centric sampling process! –  ShreevatsaR May 5 '11 at 7:32
    
The same question can be asked on Khovanova. She says: Now let me give you a variation of the Tuesday-Child problem that is unambiguous and 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 a Tuesday?”. If the answer is yes, what is the probability that the father has two sons? The 13/27 argument works perfectly in this case. Why there its unambiguous? There I also can ask "Do you have a son born on some weekday?" and he answer "Yes". The same problem exists. –  Tangent Bundle May 5 '11 at 7:58
    
If you ask "do you have a son born on some weekday?" (without naming a weekday) and the answer is "yes", the probability of two sons is unambiguously 1/3. The difference is in the questions. :-) It's clear that when a question asks for less information (and thus gets less), the posterior probability can be different. The trouble with the original statement of the question is that it confuses "X is true" with "A father says X is true" — the information in the former is precisely that X is true, whereas the information in the latter depends on what else the father could have said. –  ShreevatsaR May 5 '11 at 9:17
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This is like in the Monty Hall problem, the difference between "a door was opened that contains a goat" (makes no difference to switch) and "the host necessarily opened a door that he knew contains a goat" (it's better to switch). The space of host actions matters (or the space of father-statements), not only the bare fact that a door-with-a-goat was opened (or the bare statement that the father made). –  ShreevatsaR May 5 '11 at 9:20
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It is actually impossible to have a unique and unambiguous answer to the puzzle without explicitly articulating a probability model for how the information on gender and birthday is generated. The reason is that (1) for the problem to have a unique answer some random process is required, and (2) the answer is a function of which random model is used.

  1. The problem assumes that a unique probability can be deduced as the answer. This requires that the set of children described is chosen by a random process, otherwise the number of boys is a deterministic quantity and the probability would be 0 or 1 but with no ability to determine which is the case. More generally one can consider random processes that produce the complete set of information referenced in the problem: choose a parent, then choose what to reveal about the number, gender, and birth days of its children.

  2. The answer depends on which random process is used. If the Tuesday birth is disclosed only when there are two boys, the probability of two boys is 1. If Tuesday birth is disclosed only when there is a sister, the probability of two boys is 0. The answer could be any number between 0 or 1 depending on what process is assumed to produce the data.

There is also a linguistic question of how to interpret "one is a boy born on Tuesday". It could mean that the number of Tuesday-born males is exactly one, or at least one child.

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+1. We have a reasonably standard jargon for specifying simple random processes in English. People come up with "puzzles" like this by exploiting the impreciseness of English for describing more complicated situations. If the person posing the question had directly stated the random process, nobody would be confused. The reason this is a "puzzle" is that it's worded too vaguely to convey the intended meaning. So @muad: any time someone asks you a probability "puzzle", you should simply press them about exactly what random process they have in mind. Then it will be easy. –  Carl Mummert Sep 12 '10 at 12:07
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The Tuesday is a red herring. It's stated as a fact, thus the probability is 1. Also, it doesn't say "only one boy is born on a Tuesday". But indeed, this could be a language thing.

With 2 children you have the following possible combinations:
1. two girls
2. a boy and a girl
3. a girl and a boy
4. two boys

If at least 1 is a boy we only have to consider the last three combinations. That gives us one in three that both are boys.
The error which is often made is to consider 2. and 3. as a single combination.

edit
I find it completely counter-intuitive that the outcome is influenced by the day, and I simulated the problem for one million families with 2 kids. And lo and behold, the outcome is 12.99 in 27. I was wrong.

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Note that this too may seem counter-intuitive: It means that if someone tells you "I have two children. One of them is a boy" the probability the the other one is a girl is 2/3, but you might expect 1/2 since the sex of the children is independent. –  Gadi A Sep 11 '10 at 7:00
    
What you are saying here contradicts my derivation of 13/27 (which I cannot see any flaw in) - can you tell me if you think there is a mistake in my derivation? –  anon Sep 11 '10 at 7:03
    
@muad: there doesn't seem a mistake in your derivation; the error was in my intuition that the day couldn't possibly have anything to do with it. –  stevenvh Sep 12 '10 at 7:04
    
I also had to write a computer simulation before I could believe the answer! –  anon Sep 12 '10 at 7:49
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Well, given the unstated assumption that the writer is a mathematician and therefore not using regular english, then I agree with the 13/27 answer.

But in everyday english, from "there are two fleems, one is a glarp" we all infer that the other is not a glarp.

From "there are two fleems, one is a glarp, which is snibble" we would still infer that the other is not a glarp. Whereas from "there are two fleems, one is a glarp which is snibble" (absence of comma, or when spoken, difference in intonation) we would infer that the other is not a snibble glarp, but it could still be an unsnibble glarp.

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I see what you mean, Whether one interprets "One is ..." to mean "Exactly one (and no more) is ..." or "At least one ...". This would change the answer from 13/27 to 12/26. I don't think this ambiguity is an intentional part of the problem though - just an unfortunate problem of the language. –  anon Sep 12 '10 at 7:57
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I think "But in everyday english, from "there are two fleems, one is a glarp" we all infer that the other is not a glarp." is debatable. –  Chris Card Sep 12 '10 at 8:31
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This, in my opinion, is why the intuitive approach fails:

One has a tendency to think that the probability of 7*P(b AND d1) = P(b AND d1) + P(b AND d2) + ... + P(b AND d7) = P((b AND d1) OR (b AND d2) OR ... OR (b AND d7)) = P(b AND (d1 OR d2 OR ... OR d7)) = P(b).

However, the flaw here is that, in reality, P(b AND d1) + P(b AND d2) + ... + P(b AND d7) is NOT equal to P((b AND d1) OR (b AND d2) OR ... OR (b AND d7)). This means that mentioning independent (and one might think irrelevant) information alongside with relevant information actually changes the resulting probabilities.

One interesting consequence: if I say something like "I have two children. One of them is a boy who was born at 10:24 PM on February 10th," The probability that I have two boys is now almost exactly the same as as the probability that I have a girl and a boy. Adding a unique or almost unique piece of information makes the stuff I want to know about the other child independent of the information I have on the first child. If I took this to the extreme and said that I have a firstborn boy, won't know anything additional about the other child.

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What is an example of the "... is NOT equal ..."? (A well-specified probability model where the two sides of the equation are not the same.) –  T.. Sep 13 '10 at 1:26
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I guess the following two versions of the experiment provide two different answers:

  1. Dave has two children. Is atleast one of them a boy who is born on Tuesday? Dave answers Yes.

  2. Dave has two children. I ask him to first choose and fix one child at random, and tell me if it is a boy who was born on Tuesday. Dave answers yes he is a boy born on Tuesday.

For 1st the answer is 13/27, while the second has answer 1/2.

The way in which the question is asked, it's in line with 1st, hence the answer should be 13/27.

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+1 for correct answer. More specifically, your first example is equivalent to the case where Dave knows the sex/birthdate of both children, while the second example is equivalent to the case where Dave doesn't know the sex/birthdate of both children; so what we have here is yet another question about knowledge-of-knowledge. See this answer for more information. –  BlueRaja - Danny Pflughoeft Dec 21 '10 at 20:11
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Also, since there is - presumably - no ambiguity over whether or not Dave knows the sex of his children, there is indeed no ambiguity in the quesiton; the answer is correctly 13/27 (the accepted answer, by Gadi, is incorrect). –  BlueRaja - Danny Pflughoeft Dec 21 '10 at 20:13
    
@BlueRaja-DannyPflughoeft: The issue isn't whether Dave knows his children, but under what circumstances he made his statement. In other words if Dave says "I have two children, and at least one of them is a boy born on a Tuesday", the information conveyed by his statement depends on the space of possible ones. (You're assuming that the information conveyed is the content of his statement: and yes, the answer to "if we are given the information that Dave has a boy born on a Tuesday..." is indeed 13/27.) See Tanya Khovanova's article "Martin Gardner's Mistake". –  ShreevatsaR Jan 17 at 5:38
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There is always room for misinterpreting a question when one does not fully understand the language in which it is written. I think that the way mathematics and mathematicians use conditional probability is clear:

$$P(A|B)=P(A \cap B)/P(B).$$

So I believe that this is the interpretation that one should take, and thus arrive at your answer of 13/27, and not search for further nuances, which are not too difficult to find.

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Conditional probability is not the issue. The problem is in defining what "P" is implied in the problem. Once a probability measure is specified ,of course it is then possible to use the formula for conditional probability. –  T.. Sep 11 '10 at 19:08
    
So A = [Dave has 2 boys] and B = [Dave says "I have 2 children. One is a boy born on a Tuesday."]. The problem is that we don't know P(B). What space does Dave's statement come from? The only way to get the 13/27 answer is to make the unjustified unreasonable assumption that Dave is boy-centric & Tuesday-centric: if he has two sons born on Tue and Sun he will mention Tue; if he has a son & daughter both born on Tue he will mention the son, etc. See [this article](arxiv.org/abs/1102.0173). The information in [Dave says X] is not the same as [X]; it depends on what all Dave could have said. –  ShreevatsaR May 5 '11 at 11:08
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