## The Tuesday Birthday Problem 981

An anonymous reader sends in a mathematical puzzle introduced at the recent Gathering 4 Gardner, a convention of mathematicians, magicians, and puzzle enthusiasts held biannually in Atlanta. The Tuesday Birthday Problem is simply stated, but tends to mislead both intuitive and mathematically informed guesses.

*"I have two children, one of whom is a boy born on a Tuesday. What's the probability that my other child is a boy?"*The submitter adds, "Believe it or not, the Tuesday thing is relevant. Well, sort of. It's ambiguous."
## Re:Ordering and Convergence (Score:5, Insightful)

## Re:Let's try it without reading TFA (Score:2, Insightful)

## Other problems (Score:3, Insightful)

"I have two children, one of whom is a boy born in the first day of the year. What's the probability that my other child is a boy?"

"I have two children, one of whom is a boy born in January. What's the probability that my other child is a boy?"

"I have two children, one of whom is a boy born in Winter. What's the probability that my other child is a boy?"

Do they give different probabilities?

## Principles of Restricted Choice (Score:4, Insightful)

This is related to the Principle of Restricted Choice [wikipedia.org] often seen in Contract Bridge.

If the parent has two boys born on a Tuesday, he could equally have declared the other boy as being born on a Tuesday. In a parallel universe, the other boy would have been declared as being born on a Tuesday, whereas if only one of the child was a boy born on Tuesday nothing would have changed in any of the other parallel universes. Therefore the effect is the probability of 2 boys borne on Tuesday has been halved, resulting in 13/27 probability of the second child being a boy.

## Re:Let's try it without reading TFA (Score:2, Insightful)

You are good:) I have a question.

I met my girlfriend on Thursday. She is receptionist. What is the probability of my second girlfriend being supermodel?

## Science and Intuition defeating Fun Math (Score:1, Insightful)

Take a thousand families, with two children, where one of the children was a boy born on a Tuesday.

I don't mean a thousand theoretical families. I mean, lets say you straight up took one thousand real families, that matched the above constraints, straight out of the census. No joke, you break out the SQL.

When you check the gender of the other child, you are going to see the breakdown of gender being 50% male, 50% female.

Now, I know there's a lot of fun handwaving going on. Here's the flaw, in a nutshell. There are indeed three possibilities, when one child is constrained to be a boy:

boy, girl

girl, boy

boy, boy

The mistake -- and it is a mistake, because when you actually run the experiment, the hypothesis is invalidated -- is thinking that each of the above cases is equally likely. Specifically, order of birth has been incorrectly elevated as a determining factor. So we see:

boy, girl: 33%

girl, boy: 33%

boy, boy: 33%

When we really should be seeing:

boy, boy: 50%

boy, girl: 25%

girl, boy: 25%

Or, more accurately:

same-gender, both male: 50%

different-gender: 50%

boy first: 25%

girl first: 25%

Another way to frame the query, with similar results, is to say:

Select the gender of all second children where the first child was born on a Tuesday and the first child was male.

Select the gender of all first children where the second child was born on a Tuesday and the second child was male.

You'll note the girl, girl families will show up in neither result set. So they can do nothing to skew the numbers.

The results of both queries will, predictably, be 50/50 male and female.

This is a good example of why framing a problem correctly is so difficult and critical. It's only because this problem is so amenable to experimental formulation that it's easily defensible.

(Note that the use of Tuesday was an excellent DoS against math geeks.)

(Note also, by the way, this is the exact opposite of the Monty Hall problem. In that problem, people are expecting:

Door 2: 50% ...when, really, we have:

Door 3: 50%

Host Told You Where The Car Was: 66%

Was Behind 3, Therefore Exposed 2: 33%

Was Behind 2, Therefore Exposed 3: 33%

Host Didn't Tell You Where The Car Was: 33%

Randomly Exposed 2: 16.5%

Randomly Exposed 3: 16.5%

If you modify the Monty Hall problem, such that he opens a random door *which might actually expose the car*, then when he opens the door and you see a goat, it doesn't matter whether you switch or not.)

## The other problem posed in TFA (Score:1, Insightful)

Suppose that Mr. Smith has two children, at least one of whom is a son. What is the probability both children are boys?

This is the question posed without the birth weekday specified. TFA actually tries to say that there are 4 outcomes for the

pair of children, one of which is impossible, so they remove it. Since "boy, boy" is only one of the 3 outcomes, then the probability must be 1/3. Right?Wrong.The boy (let's call him Peter) being a

boyis agivenof the problem, so it has P = 1. The other child -- we don't care about it being born before or after Peter -- is independent, so the probability that it's a boy is 0.5*. The 4 outcomes are as follows:Peter, Boy = 0.25*

Peter, Girl = 0.25*

Boy, Peter = 0.25*

Girl, Peter = 0.25*

So, whichever way we slice this problem, the solution is 0.5*.

P(Peter, Boy) + P(Boy, Peter) = 0.5*

1 * P(Other is Boy) = 0.5*

- - - - - -

* May slightly differ due to the male:female ratio at birth. It is assumed here to be 1:1.

## Re:Summary misstates the problem (Score:5, Insightful)

"I have two children, one of whom is a boy. What's the probability that my other child is a boy?" ... it is given that the FIRST child is a boy.

I must admit that English is not my native tongue but I fail to see how this gives that the FIRST child is a boy. Doesn't "one of whom" implies that it can be either the first or the second?

## Re:Rubbish (Score:2, Insightful)

## Re:What's counterintuitive about it? (Score:4, Insightful)

This is a question written in purposely misleading English.

This, in other words, is a shit question.

## Interesting contrivance of math and grammar (Score:2, Insightful)

## Re:Rubbish (Score:3, Insightful)

I'm at a company picnic and I have 5 employees sitting around. I need 2 more players for the women's volleyball team, so I take 2 of the women away. What is the sex distribution of the rest? The issue is I'm cherrypicking based on a condition. Here the condition is clear: I'm picking based on sex.

The confusion in the Tuesday problem comes in because the condition doesn't appear relevant. Who cares about Tuesday?? The assumption is he picked one of his kids to talk about.

## Re:Ordering and Convergence (Score:5, Insightful)

Normal English strongly implies it does. If you say to someone I have several pieces of fruit and one of them is a banana, when in fact two of them are bananas, most people would call that lying. One could argue that strictly speaking the statement was true: you did have one banana, you just also had an additional banana, but that level of honesty is only tolerated in politicians. If you had said "at least one of which is a banana" that would be fine, otherwise the statement is deliberately misleading.

## Re:0.5 (Score:2, Insightful)

## Re:Ordering and Convergence (Score:5, Insightful)

Not necessarily. Any reading of an English sentence is an exercise in interpretation. We don't just read the words alone, we actually interpret them and use a mental model to help dismiss obviously incorrect ambiguities.

Let's say the sentence has several interpretations. For each interpretation, we could solve for an interpreted probability answer. Then we could look at each answer and ask if it makes sense. If that answer doesn't make sense, we could dismiss that particular interpretation of the sentence. If a single answer remains after that, it would be THE answer (Sherlock Holmes style).

In the example, here are some possible disambiguations:

1) "I have two children, (exactly) one of whom is a boy born on a Tuesday."

2) "I have two children, (at least) one of whom is a boy born on a Tuesday."

3) "I have two children, one of whom is a boy born on a (particular) Tuesday."

4) "I have two children, one of whom is a boy born on a (generic) Tuesday."

There is also an ambiguity in the second sentence, which is only obvious to statisticians and probabilists:

a) "What's the (Bayesian subjective) probability that my other child is a boy?"

b) "What's the (objective) probability that my other child is a boy?"

In case a), the problem is underspecified as it requires the full set of personal beliefs of the reader to be used for an answer (Bayesian subjectivists propose that a probability is merely a degree of personal belief, such that two people will not agree on the probability for the same event, because, being different people, they have different prior beliefs.)

In case b), the problem can (should) be solved solely from the problem and general common knowledge of the world (which is still required to interpret the question).

## Re:Ordering and Convergence (Score:2, Insightful)

These kind of problems require competence in both English and Maths which is why so few people get them right.

The number "one" applies to the clause (boy born on tuesday) so the day of the week is relevant.

If you say to someone I have several pieces of fruit and one of them is a banana, when in fact two of them are bananas, most people would call that lying. One could argue that strictly speaking the statement was true: you did have one banana, you just also had an additional banana, but that level of honesty is only tolerated in politicians. If you had said "at least one of which is a banana" that would be fine, otherwise the statement is deliberately misleading.

## Man this question pisses me off. (Score:5, Insightful)

"I have two children, one of whom is a son born on a Tuesday. What is the probability that I have two boys?"

I have three children. But I also have 2 children. See the problem? I have a son born on a Tuesday, and I have another son born on a Tuesday. See the problem?

It doesn't say I have *only* two children. It doesn't say the other child can't be a son born on a Tuesday. It assumes the birth rate is 50/50, but most statistics agree it's not even. FTA, it assumes there's no such thing as twins. It assumes you have only one wife. But none of this shit is specified.

Pisses me off. Use coins and cards. Not assumed biblical customs.

## Re:Ordering and Convergence (Score:5, Insightful)

Yes, it is like the Monty Hall problem. What is the probability that 2 children are both boys? 25%. Knowing that one of the children is a boy does not change that probability as much as might be thought. The answer then is 33.3%, not 50%. This is because the additional information has cleverly NOT specified which child is the boy. If a particular child is picked out, eg. the

firstchild is a boy, then it is 50% the other is a boy, because it always was 50% likely that a child is a boy. The bit about "born on Tuesday" does matter, because it comes close to specifying a particular child. The more improbable it is that both children fit some criteria, the closer the probability gets to 50%. If the info had been "one of whom is a boy born on Feb 29", the answer would be nearly 50%.## Re:Science and Intuition defeating Fun Math (Score:3, Insightful)

#!/usr/bin/perlmy @set;for my $gen (1 .. 100000) {my $sex1 = rand > 0.5 ? "m" : "f";

my $sex2 = rand > 0.5 ? "m" : "f";

push @set, [$sex1, $sex2];}

my $count = 0;my $total = 0;

foreach my $pair (@set) {next if ($$pair[0] ne "m" and $$pair[1] ne "m");

$total++;$count++ if ($$pair[0] eq $$pair[1]);}

print "$count / $total\n";## Re:Well? (Score:3, Insightful)

Here is the list of outcomes if it is possible for both boys to have been born on a Tuesday:

Girl on Monday, Boy on Tuesday

Girl on Tuesday, Boy on Tuesday

Girl on Wednesday Boy on Tuesday

Girl on Thursday, Boy on Tuesday

Girl on Friday, Boy on Tuesday

Girl on Saturday, Boy on Tuesday

Girl on Sunday, Boy on Tuesday

Boy on Monday, Boy on Tuesday

Boy on Tuesday, Boy on Tuesday

Boy on Wednesday Boy on Tuesday

Boy on Thursday, Boy on Tuesday

Boy on Friday, Boy on Tuesday

Boy on Saturday, Boy on Tuesday

Boy on Sunday, Boy on Tuesday

If having a boy or girl is equally likely (we dont do any weighting), then the chance its a boy is therefore 7/14=1/2. If you couldnt have two boys on Tuesday it would be 6/14=3/7. Please point out where this analysis goes astray.

## Re:The other problem posed in TFA (Score:4, Insightful)

In 1000 pairs of children you'll have 250 girl/girl, 250 boy/boy, 500 girl/boy.

Of those, the ones that have at least one boy are the 250 boy/boy and 500 girl/boy pairs. So there's a 33% chance it's boy/boy if you know one is a boy.

The whole point is you could be talking about either of the boys in the 250 boy/boy pairs - it doesn't increase the probability that it's boy/boy instead of girl/boy (you're still twice as likely to have a girl/boy pair relative to a boy/boy pair). If you specify more about the boy you're talking about - for example (ironically) saying his name is Peter - then the boys are no longer interchangable and the probability tends towards 1/2.

It is tricky

## Re:Well? (Score:4, Insightful)

My older brother and I were both born on Tuesdays.

## Re:Rubbish (Score:2, Insightful)

But yet we interpret the "two children" as meaning exactly two.

"two children" is an unambiguous statement ... it can't mean one child, it can't meant three children, neither can it mean two dogs.

"one of whom" can be ambiguous ... it can mean only one (of the children), or just the one I am describing. Nowhere in the original statement is it said that the second child was not a boy born on a Tuesday. You can argue it's implied, but it's not stated.

Just because you say that the first child was a boy born on a Tuesday doesn't mean that the second can't be the way the statement is worded. This is a mathematician using English badly to prove his point!

## Re:I love mathematicians... (Score:3, Insightful)

So the correct answer is "That depends. Is your other child a boy?"

## Re:Well? (Score:2, Insightful)

IANA mathematician, but: if you also list the probabilities for the Boy on Tuesday being the first child, you get 28 possibilities, of which 14 have two boys (giving you 1/2 again). However, then you've listed "Boy on Tuesday, Boy on Tuesday" twice, although there's no reason for it to be more likely than any of the other possibilities. So if you remove the duplicate, you get 13/27, as stated.

If, on the other hand, it stated that the younger child was a Boy born on a Tuesday, your list would apply, so the probability of two boys would be 1/2.

## Re:Man this question pisses me off. (Score:2, Insightful)

## As always, ambiguous language (Score:3, Insightful)

"I have two children, one of whom is a son born on a Tuesday. What is the probability that I have two boys?"

As always, the challenge is the assumptions intentionally hidden in the problem statement.

"I" - was your family chosen at random, and if so, from what set?

"two children" - exactly or at least?

"one of whom" - exactly or at least?

"son" - was the sex to say chosen at random, or did you pick a child and announce his/her sex?

"Tuesday" - was the day chosen at random, or did you pick a child and announce his.her birthday?

"What is the probability..." - Some parent you are! Don't you know the sex of your own children?

Simply and honestly reveal the assumptions and the math is straightforward.

"Given a family, chosen at random from the set of all families that have exactly two children and have at least one son born on a Tuesday, what is the probability that both children are boys?"

To make the math easier, let's start with 196 families with two children, with the expected mix of boys and girls. 49 (25%) have two boys and 98 (50%) have a boy and a girl. Of the 98 boy-girl families, 84 do not have a Tuesday-Boy, leaving 14 that do. Of the 49 boy-boy families, 36 do not have a Tuesday-Boy, leaving 13 that do. That leaves a total of 27 families, of which 13 have at least one son born on a Tuesday.

So the probability is 13/27.

Reveal different assumptions, and the answer changes.

## Re:Ordering and Convergence (Score:3, Insightful)

This problem hinges very greatly on how it is phrased and I think it's more a trick of English converting to statistics than it is a true puzzle.

100% correct. All "paradoxes" of this form are very carefully, almost ritually, stated because they are almost entirely linguistic tricks designed to mislead and confuse rather than educate and enlighten.

The very first thing anyone analyzing a problem of this type should do is restate it in several different ways, making as much implicit information explicit as possible. Such as:

I have two children.

I'm going to tell you some things about one of my children.

I have not chosen that child randomly. For example, I am always going to tell you about my male child. I have no idea what people do when they don't have male children. The very fact that I'm telling you this puzzle means I must have at least one male child, since this puzzle is always stated in terms of male children.

The child I have non-randomly decided to tell you about was born on a particular day of the week.

His name is Fred, by the way. Cute kid.

The child I have non-randomly decided to tell you about is also male, but then again, we established that by the very fact of my telling you this.

Now that I've told you all those non-random things about one child, I want you to assume total randomness and make what I think is the "correct" inference about another child. You're almost certain to get it wrong, and then I'm going to laugh at you for not treating everything as totally random after I've fed you a bunch of carefully chosen, non-random information.

An honest mathematician would of course select a few dozen facts about thier children and then dice to get two of them about one of the children, and present the question in those terms. Then one could legitmately assume randomness in the analysis.

## Re:Ordering and Convergence (Score:5, Insightful)

These kind of problems require competence in both English and Maths which is why so few people get them right.

Mostly they require competency in psychology, so you can figure out how the twit posing the problem is deliberately trying to mislead you by using ambiguous English and claiming on the basis of their poor communication skills to be clever.

## Re:Ordering and Convergence (Score:2, Insightful)

Normal English strongly implies it does. If you say to someone I have several pieces of fruit and one of them is a banana, when in fact two of them are bananas, most people would call that lying. One could argue that strictly speaking the statement was true: you did have one banana, you just also had an additional banana, but that level of honesty is only tolerated in politicians. If you had said "at least one of which is a banana" that would be fine, otherwise the statement is deliberately misleading.

The problem is, nobody would say anything like this in "Normal English." Normal people do not go about asking about the probabilities that one of your children is male or female in regular conversation; they just say, "I have a son and a daughter," (or "two sons," or whatever.)

Likewise, if someone's not telling you what kind of fruit they have and only admits to having one banana, you start to suspect that they're hiding a truckload of moldy bananas somewhere and you should probably not be buying any fruit from that person.

## Re:Well? (Score:2, Insightful)

But if you split one of those cases in two, each has only half the chance of occuring in the first place.## Re:Ordering and Convergence (Score:1, Insightful)

Normal English strongly implies it does. If you say to someone I have several pieces of fruit and one of them is a banana, when in fact two of them are bananas, most people would call that lying. One could argue that strictly speaking the statement was true: you did have one banana, you just also had an additional banana, but that level of honesty is only tolerated in politicians. If you had said "at least one of which is a banana" that would be fine, otherwise the statement is deliberately misleading.

I've taken too many tests where the wording is intentionally ambiguous in this way, in order to make sure the student pays close attention to the exact words used and answers the question asked. There are oftentimes additional, not necessarily relevant facts included in the question. Now, in normal conversation, I can see this being deceptive (I hear lawyers get paid well for such tactics) but this appears to be an academic question.

"I have two children, one of whom is a boy born on a Tuesday. What's the probability that my other child is a boy?"

This gives one general fact: "I have two children"

It gives a specific fact about one of them, which may or may not have anything to do with the other: "one of whom is a boy"

It gives another specific fact about that same one, which may or may not have anything to do with the other: "[who] was born on a Tuesday."

It then asks a question: "What's the probability that my other child is a boy?"

As far as my [perhaps flawed] reasoning can see, the question could simply be, "What's the probability that a child is a boy?", that is, any child.

## Re:MOD PARENT UP (Score:5, Insightful)

In my experience, many non-intutive probabilty results are easier to understand if you spell out the full population. For example, I coudn't understand http://en.wikipedia.org/wiki/Berkson's_paradox [wikipedia.org] until I drawed it up graphically.

## Re:Well? (Score:1, Insightful)

## Two cases of Tuesday boys (Score:1, Insightful)

The AC you responded to has it right, both cases of "two Tuesday boys" need to be counted. You are also correct that, given two boys born on Tuesday, there is equal probability that a specific one is older or younger than the other. This second fact, however, is not a good reason for excluding one of the cases. In fact, it's the precise reason why we need to count both.

Here are the two "two Tuesday boy" cases as I see them:

(Boy I've met is older, boy I haven't met is born on Tuesday)

(Boy I've met is younger, boy I haven't met is born on Tuesday)

Excluding the second of these instances is equivalent to saying that if I'd known the boy I met was the younger then I'd also know that his brother couldn't have been born on Tuesday. This is clearly not a true claim.

Neither would it be reasonable to exclude the first one, for the same reason: knowing that the boy I've met is older than his brother doesn't mean that his brother couldn't be born on Tuesday, too.

The probability of the "brother I haven't met" being older or younger must be uniform, which requires two equally likely cases to be counted.

## Re:The other problem posed in TFA (Score:1, Insightful)

You're not listening to what he's saying. The selection of the unknown child in the example given isn't realistic based on the assumptions a normal person would have when posed with that question. Let's replace the children with colored balls in a bag. You were told they can be either green or red, therefore the probability of one of the balls being red is 1/2. If you pull a ball out, look at it and confirm at least one of them is red, the probability of the second ball being red is still 1/2. Now, if you take that red ball you pulled from the bag AND PUT IT BACK...what's the probability you will pull out a red ball now? 1/3.

"If one of my two children is a boy, what's the probability the other is a boy?" is 1/2

"If one of my two children is a boy, what's the probability the one YOU CHOOSE AT RANDOM will be a boy?" is 1/3

## Re:Ordering and Convergence (Score:3, Insightful)

This problem (and related problems) are almost always poorly phrased. However, your assertion that failure to ask about the day-of-week of the 2nd child's birth makes the day-of-week of the 1st child's birth irrelevant is incorrect, if the problem is understood as it was intended. And actually my own phrasing there is a bit poor there, because the crux of the problem is that there is no ordering - no "1st child" and "2nd child".

A better (or in any event less misleading) phrasing would be: "There are two children. It is a given that one of them is a male born on Tuesday. What are the odds that both are males?"

IMO the most explicit way to state the problem is: Take a large number of 2-child families. Dismiss any of them that do not have at least one male child who was born on a Tuesday. Randomly select one of the remaining families. What are the odds that the family you've selected has two male children?

In that phrasing, it's easier to see why the answer is less than 50% (since I didn't distinguish one child from the other you can't treat their genders as independent events) but more than 25% (the demographics of the families you dismiss are skewed). The "birthday" factor changes the skew, so it is not irrelevant.

It is a common complaint among those who don't reach the correct answer, that this isn't really the question that was asked. It is; it was just asked poorly.