The Tuesday boy problem, in under 300 words
mikeschiraldi.blogspot.ca
mikeschiraldi.blogspot.ca
http://scienceblogs.com/evolutionblog/2011/11/08/the-tuesday...
The attempt at shortening it down to under 300 words omits all the complications and subtlety involved in the problem.
The one that occurred to me is that the probability of getting a boy or girl is not 50%, and the probability of getting two same-gender children is not 50% either. Here are some nice real stats: http://www.in-gender.com/XYU/Odds/Gender_Odds.aspx
In maths exams at school the problems are always presented in the form of: "a random family was chosen from ...". (Similar to "a smooth cube is resting on a smooth surface".
In real life, it's more difficult. The original article looks at real life, whereas the 300-word version has reduced it to the exam version. Of course it's "easier" to understand the exam version, but this wasn't the point.
From archgoon’s link:
> "This puzzle confounds people legitimately, however, because most of the ways in which you are likely to find out that X has at least one boy contain an implicit bias which changes the answer.”
Exactly right. The phrasing in the problem asks the reader to consider a particular scenario, not isolated facts. Consider what happens if you take the question at its word:
> If a person on the street says to you, "I have two children," what is the probability that they're both boys?
Hell if I know. Maybe one is a boy and the other is a grown man? Or maybe he has five children, BBGGG. “I have two children, both of which are boys” is technically correct if there’s two boys and three girls, right? Or maybe he’s just outright lying! How do I assign a probability to any of those?
Consider this phrasing instead:
“If you flip a coin twice, and the first and/or the second is heads, what’s the probability that both are heads?” I’d expect more people to get this right than the boy/girl question.
Also, as an aside, punctuation is important:
“I have two children. At least, one of them is a boy.”
(Yikes!)
This removes the whole ambiguity of why he would assert "at least one of them is a boy," but still makes the question a bit tricky/counter-intuitive, which is useful for teaching the quirks of probability.
In the case of BB, the stranger will always say, "At least one is a boy." Given GB or BG, they have even odds of saying "at least one is a boy" vs "at least one is a girl." Given GG, the stranger will always say, "At least one is a girl." The correct diagram would be this: http://abughrai.be/pics/boy_probability.png
For the probabilities in that post to apply, the stranger would say, "I have two children." Then you ask, "Is at least one of them a boy?" and they answer, "Yes."
Edit: Of course, the answer to, "Given there are two siblings and one is a boy, what are the chances both are boys?" is 1/3. But the phrasing of the puzzle does not reflect that question.
Rather, what they said is given, and the question is: what pair of children do they have?
You are either going over my head or under my head, so this will probably make one of us temporarily look a bit foolish :).
Suppose the puzzle is phrased this way: X says “I have two children and at least one is a boy.” What is the probability that the other is a boy?
Put this way, the puzzle is highly ambiguous. Computer scientists, cryptologists and others who must deal carefully with message-passing know that what counts is not what a person says (even if she is known never to lie) but under what circumstances would she have said it.
The way the puzzle is phrased, the only way to get 1/3 as an answer is to assume that parents with one girl will never say, "At least one of them is a girl."
1. http://scienceblogs.com/evolutionblog/2011/11/08/the-tuesday...
> The way the puzzle is phrased, the only way to get 1/3 as an answer is to assume that parents with one girl will never say, "At least one of them is a girl."
Right. But that is assumed, because the puzzle states that the parent says, "At least one of them is a boy."
In other words, "Put this way, the puzzle is highly ambiguous." No, not it is not. It is absolutely not ambiguous.
I made this point in a different comment for a different blog post with the same example, but: both things you link to contain a fallacy, as follows.
"Suppose we know that a certain man has two children and we also know that the older one is a boy" IMPLIES "Now suppose we know simply that a man has two children and that one of them is a son. This time we would reason that there is no possibility that the person has two girls. It follows that the sexes of his two children, ordered from oldest to youngest, are either BB, BG or GB...."
You can't conclude, as these authors do, from the first case, that the other child's gender is 1/2, and from the second, it is 1/3. It's not that the two cases are different (in any important way); they are not; one implies the other.
It's that in the 1/2 case, the person is naive, and in the 1/3 case, the person takes into account the known fact that there is a 1/2 chance per child of getting a certain gender, thus allowing the BB/BG/GB/GG analysis.
You can't solve a logic puzzle by arguing the psychology of a persona depicted in the puzzle.
GG GB BB
The relative ages of the children play no part, they could be twins for all I care. So the initial probabilty of being BB is 1/3. Knowing one of them is a boy then removes GG as an option, and the probability for BB unsurprisingly becomes 1/2.
What am I misunderstanding in this problem?
The first child can be boy or girl. The second child can also be boy or girl. So the probabilities are like this: BB: 25% GG: 25% BG: 50%
You can replace older/younger on the article with child 1/child 2.
lim[generic -> specific] f(P("is one child of two a boy")) = 1/2
Obviously I have no proof for it but it is an interesting observation.
Intuition wrongly treats the possibilities as sets and assigns probabilities 1/2 BB, 1/2 GB, rather than 1/3 BB, 1/3 GB, 1/3 BG.
See also http://lesswrong.com/lw/oj/probability_is_in_the_mind/
I responded—note that this was completely spontaneous—"What on Earth do you mean? You can't avoid assigning a probability to the mathematician making one statement or another. You're just assuming the probability is 1, and that's unjustified."
While the equations of probability theory are agreed upon, there are multiple interpretations[1]. In some ways, it's like QM.[2]
1. http://en.wikipedia.org/wiki/Probability_interpretations#Phi...
2. http://en.wikipedia.org/wiki/Interpretations_of_quantum_mech...
> But suppose that instead you had asked, "Is your eldest child a boy?" and the mathematician had answered "Yes." Then the probability of the mathematician having two boys would be 1/2. Since the eldest child is a boy, and the younger child can be anything it pleases.
No, it cannot be "anything it pleases." That is the whole point of the blog post that we are all commenting about here on HN, which is re-told on the blog post you are linking to.
Since the post you are linking to is misrepresenting one side of the "argument," I strongly suggest that there is no actual apparentl contradiction here that needs to be resolved.
Yes, you can talk about the actual probability of something vs. making the best possible prediction you have. A good example raised in this post is the point about a biased coin; you know it is biased but not which way, so you assign odds of .5 for heads and .5 for tails. Fine. This is not some massive philosophical conundrum.
The fact is that actual knowledge is neither purely "out there" (e.g. "in the cards") nor purely "in here" (in the mind). This is the classic rationalism vs. empiricism debate, which was settled by Ayn Rand's theory of objective knowledge, where she answers that knowledge is in man's mind, but it is knowledge of external reality. So, "both."
Not understanding this, and other basic epistemological errors, seem to plague lesswrong.com. It is embarassing to see them supposedly championing "reason" under a non-objective epistemology. They come to bizarre conclusions, and that is, at least partially, why.
(1) Do you hold that concepts and principles are purely "in" the referents?
(2) Do you hold that they are purely "in" man's mind?
(3) Do you hold that they are "in" man's mind, but refer to properties of the referents? (Rand's conclusion)
(4) Do you hold that Rand's argument and conclusion is correct, but somebody else made it earlier?
(5) Do you hold that there is insufficient evidence to answer the question (which then raises questions about why you would even engage in an intellectual discussion)? For example, you could "argue" for a subjectivist view of knowledge.
An actual answer along the lines of 1-5 would be useful, but stating "I don't think so" does not provide any reasoning, and thus is utterly devoid of intellectual content.
You seem to be dismissing a theory without considering and responding to its evidence, which is fraudulent science. Nobody wants to see that kind of "argument," even people who nonetheless do agree with the conclusion.
I do think you have the intellectual right to merely state that you disagree, but you have to say that you don't intend to defend it, or let yourself be subject to having it pointed out by someone, as I am.
That is because too many people today hide behind the assumption that agreement or disagreement without evidence presented is enough to establish or disestablish truth. As I said before, that is fraudulent science.
In oversimplifying your table for potential idiot readers by introducing the silent variable of older vs younger child, you have introduced order into a statistics problem wherein order is irrelevant and masked the fact that what matters is which of the children you already know to be a boy. You grant this difference for the BG pairing, thus producing two options, but not for the BB pairing. There is not ONE way of knowing this for BB, as you present in the table, but two: you know the older is the boy, or you know the younger is the boy. 2/4 = 1/2. QED.
What you should have done in the first place, however (unless your aim is to produce a blog post which, apparently, can convince otherwise intelligent people that irrelevant information can magically become relevant) was simply remove the one boy from the equation and reformulate the question. What is the probability that this other child is a boy (and thus that both are boys)? 1/2. Fuck the table. Had you not used a table, this would never have happened. But it feels authoritative, right? Thank you for the psychology lesson.
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Downvotes but not refutations, because there aren't any. I'll assume it's my tone. My opinion of the competence of the average reader on this site has plummeted reading the other comments, however, so hey, fuck you guys too. :)
The problem is the interpretation of the question in precise terms. We are given some background information and then told the observation of a particular random variable. In the given article, the random variable is interpreted as "yes/no: at least one child is a boy born on a Tuesday". Rambling about "order" does not change the fact that the article is correct, given the definition of the random variable that is observed.
The problem is that this interpretation is only correct if you imagine that you are always given the observation of the same random variable no matter what the genders or birthdays of the children actually are. For example, if I had two girls born on Wednesday, I would then tell you "no child is a boy born on a Tuesday". That's an unlikely scenario which is what makes it legitimately surprising.
It's like the Monty Hall problem. What if the host picks the door in advance? What if the host only opens a door with a goat under the condition that you picked the door with the prize? What if the host only opens a door with a goat under the condition that you picked the wrong door?
The article has problems but "wrong" is not one of them.
> so hey, fuck you guys too
Hacker News is a terrible place to visit if you want to take the voting system personally. Don't make this about you. It's about the post.
You can't give a correct answer to an incomplete question. Sit down and define the problem and if you come up with a different answer it's because you made a different assumption about the definition of the observed random variable.
Let's suppose there are four universes.
* Two boys (25% chance)
* Two girls (25% chance)
* Boy and girl (50% chance)
If we assume that the observed random variable has a preimage that contains all possibilities where the statement is true, then we come to the 1/3 conclusion, which is counterintuitive but true. This is the position of the article. I have come to the same conclusion without using order, but I have used the same definition of the observed random variable.
Tell me what you think the random variable is.
We could assume that the random variable has a preimage which only contains the case where both children are boys. Bam, probability 100%, not 1/2 or 1/3. Or we can assome that the random variable has a preimage which only contains the case where one child is of each gender. Bam. Probability 0%. So any answer between 0% and 100% is defensible.
You misunderstand Monty Hall. I would know; I learned about it by being asked the question and getting it right, not by reading someone else's explanation (which leaves room for misinterpretation). The key to Monty Hall is none of what you stated (or rather, rhetorically asked); it's that the host must pick a door that is both a) bad and b) not yours. If your door is good, then he can leave either, but if your door is bad, then he must leave the good one. So, two possibilities: the door he left was either bad or good, BUT the rule driving his choice wasn't 50/50, so the odds of it being bad or good aren't 50/50, so neither are the odds of yours, even though there are only two left. Take a minute, think about it.
Now ask yourself which "door" (read child) the father told you was a boy. There's the gotcha. Under your/the author's formulation of the question, he doesn't remove a door in the BB universe, but he does in the BG universe. There are two BG's because the table acknowledges that he can tell you which of the two is definitely a boy, and thus he has two options for doing so BUT HE DOESN'T, HE HAS ONE BUT THERE ARE TWO BG UNIVERSES, SO IT LOOKS LIKE TWO.
I have in my hand a device that lets me call up on demand most of the knowledge of the world, and I use it to look at pictures of cats, and to argue with strangers.
Deleting the second option ...
No, his formulation is complete and inevitably leads to the correct answer 1/3. You are constantly avoiding a complete formulation of the problem by diverting to a muddled discussion of an entirely different problem. This is NOT about Monty Hall.
Or you're trolling. In which case you are doing it masterfully.
As I see it, there are three options - either it's a boy-boy pair, a girl-girl pair or a boy-girl pair. Now, since OP introduced the age variable, let's give the kids names to make things easier to understand. Let's say that the boys are named John and Jack, the girls Jill and Judy and if it's a boy-girl pair then they're named Jack and Jill.
Now if it's a boy-girl pair, we have exactly two options: either Jack is older than Jill or Jill is older than Jack. This is shown in the first picture.
However, we also have two options for boy-boy and girl-girl pairs: Either Jack is older than John or John is older than Jack. Either Jill is older than Judy or Judy is older than Jill. This is not shown.
Since we know that one of them is a boy, we can dismiss both girl-girl pairs. Now we're left with four options. In 50% of these options the other child is a boy.
What am I missing?
Yes, but those three options are not equally likely.
Let's say the parents decide that their firstborn will be named Jack if male and Jill if female, and the second child will be John if male and Judy if female. Now you see that there are four possibilities: Jack+John, Jack+Judy, Jill+John, Jill+Judy. The last one is eliminated when you know that there is one boy, which gives 1/3 chance that it's the first.
But then you _also_ introduced age. Because you now have two different ways of labelling the children, you've introduced a new variable, and that changes the probability.
A better way to look at it is with two unisex names: Drew and Sam. This completely replaces the age thing as a way of identifying them, so forget all about age.
Our possibilities are:
Drew: boy, Sam: boy
Drew: boy, Sam: girl
Drew: girl, Sam: boy
Drew: girl, Sam: girl
Drew: boy, Sam: boy
Drew: boy, Sam: girl
Drew: girl, Sam: boy
Consider ...
Assuming the sex of a child is 50:50, then consider all couples with exactly two children. Of those, consider only those who can truthfully say "at least one of my children is a boy." Of those couples, 1/3 will have two boys.
Interpreting the initial problem in this manner, the answer of 1/3 is right. What I'd like to know is why you think the initial problem should be interpreted in any different way.
The reason for having BG and GB is the fact that it's twice as likely to occur than each of GG and BB.
In GB and BG, he has only one way of telling you which is the boy. In BB, he has two. Thus, four possibilities a) GB, b) BG, c) BB and he gave you the first, d) BB and he gave you the second.
Your brain will of course refuse to process the above due to your earlier remark and the sweet, sweet motherfucking irony it would mean.
Not recognizing the difference is what lets some poker players make money.
Consider this question. A chap flips a coin in the morning and thereby chooses one of his two children. Today he will only talk about that child. Later I meet him and he talks about his son. I thereby deduce that he has at least one boy. What are the chances he has two sons? (corrected in edit - thank you)
Answer: 1/2.
Another version: I meet a man and his son, and in conversation it emerges that he has two children. I can see he has at least one son, what are the chances he has two sons?
Answer: 1/2.
Another version: I meet a man and it emerges in conversation that he has two children. I observe that he is carrying a shopping bag with a child's dress in it, from which I conclude that he has at least one girl. What are the chances he has two girls?
Answer: 1/3. (I had to change gender because I couldn't think of anything that would be specific to a boy.)
Your dismissal of the idea that there are other interpretations than your own is disappointing. The language in which you couch your dismissals even more so. Are you like this in person?
Do you mean "what are the chances he has two sons, not children?
By the way, thanks for your input on this thread, I'm sure I speak for many in saying that I really appreciate it!
> Do you mean ... "sons," not "children"?
And thanks for the kind comment.
The two are independent variables, man. I don't know how to explain it any more clearly than that. What's tripping people up is the idea that removing GG from the possibilities doesn't affect the local probability of either particular child being girl. It does. You must reformulate. THAT's the counterintuitive part. It's just hella hard to explain, because there are too many 2's and 1/2's floating around the problem, so when you say one cancels another, people think you're not incorporating the first to begin with, so you must be one of the dolts the author is trying to learn some knowledge.
The following, you can take or leave: probability is my thing. I got Monty Hall right when I was in 6th grade. I was an AIME invitee, if you know what that is. I got 800 on the math SAT; 36 on the math ACT in freshman year. None of these things mean anything, logically speaking, I know. I say them only to encourage you (and only you, because you're putting in effort) to consider what I'm saying as fully as you're considering what the crowd is saying.
> No. In person I can see what the other party is thinking.
> The two are independent variables, man. I don't know
> how to explain it any more clearly than that.
> What's tripping people up is the idea that removing GG
> from the possibilities doesn't affect the local probability
> of either particular child being girl. It does. You must
> reformulate.
> It's just hella hard to explain, ...
> ... you must be one of the dolts ...
> The following, you can take or leave: probability is my
> thing. I got Monty Hall right when I was in 6th grade.
> I was an AIME invitee, if you know what that is. I got
> 800 on the math SAT; 36 on the math ACT in freshman year.
> None of these things mean anything, logically speaking,
> I know. I say them only to encourage you (and only you,
> because you're putting in effort) to consider what I'm
> saying as fully as you're considering what the crowd is
> saying.
I mention these things only to point out that you might consider that you are wrong.
So, do you answer 1/3 for each of A, B, and C?
>"The two" ?? What "two"? Your "explanations" are very unclear.
The only two there are: the two children. The sex of one is independent from the sex of the other.
>You will find many things difficult if you don't learn how to slow down, take time, be precise, and explain clearly. Be more like Feynman - take time to explain things clearly and precisely.
Okay, I'll try, good buddy. Monty Hall is different because it precludes upfront the possibility of there being any ratio other than 1:2. If we were, however, to say that any door could have either a goat or a car, then Monty's elimination of some other door would not inform you on the odds of your door YOUR door. Yea or nay?
The only way for Monty to give you any information at all about whether to switch is if the rule binds him to tell you something about the odds of the unchosen doors. In the original problem, this happens, because the ratio is bound 1:2. In our revised problem, this does not happen - unless he tells you he can't remove a door because neither is a goat. But! Using this as an analogue to our children problem, we've already precluded that possibility upfront by eliminating GG, thus Monty can't provide us any information because his B-G choice is unbound by any rule.
> Why were you running a "computer simulation" of the
> problem? Were you, even?
Moving on ...
>> "The two" ?? What "two"? Your "explanations" are
>> very unclear.
> The only two there are: the two children. The sex
> of one is independent from the sex of the other.
> Monty Hall is different because it precludes upfront
> the possibility of there being any ratio other than
> 1:2. If we were, however, to say that any door could
> have either a goat or a car, then it wouldn't matter
> whether you stayed or switched. Monty's elimination
> of some other door would not inform you on the odds
> of your door YOUR door. Yea or nay?
> The only way for Monty to give you any information at
> all about whether to switch is if the rule binds him
> to tell you something about the odds of the unchosen
> doors.
> In the original problem, this happens, because the
> ratio is bound 1:2.
> In our revised problem, this does not happen - unless
> he tells you he can't remove a door because neither
> is a goat.
> Using this as an analogue to our children problem,
> we've already precluded that possibility upfront
> by eliminating GG, thus Monty can't provide us any
> information because his B-G choice is unbound by
> any rule.
If so, how? Just as Monty's choice gives no information about the door we have selected (and which therefore remains as chance 1/3 of being the winning door), telling us that it's not the case that both children are girls gives us no further information about the other three equally likely possibilities.
So in https://news.ycombinator.com/item?id=7022615 I asked you this ...
A: So now let's take another experiment. I tell
you I'm going to flip two coins until at least
one of them is heads. I do that. Now I ask you
to bet on whether or not there is a tail. What
do you think are fair odds?
B: ...
C: ...
There are three options.
1: Your answers to A, B, and C are not all the same;
2: They are all the same, and not 1/3;
3: They are all the same, and all 1/3.
> The answer to all 3 is 1/2.
I misread A. I agree that it's 1/3 if you flip pairs until at least one is heads. I disagree that living in a world where pairs of tails don't exist and living in a world where pairs of tails do exist but are ignored are the same thing.
>Are you somehow suggesting that the probability distribution of the other options is changed by this?
Yes. I'm saying it's an unwitting double Monty Hall. The fact that he can't give you 3 cars changes the odds. Going (invisibly) from 4 possible worlds (including GG) to 3 changes the prob dist of the remaining 3, by which I mean 2. If Monty must guarantee you at least one boy, then there aren't really 3 remaining possibilities. The unguaranteed can be a boy, or it can be a girl. What is the third?
>> The father says that at least one child is a boy.
>> That eliminates the option of GG. Are you somehow
>> suggesting that the probability distribution of
>> the other options is changed by this?
> Yes. ... Going (invisibly) from 4 possible worlds
> (including GG) to 3 changes the prob dist of the
> remaining 3, by which I mean 2. If Monty must
> guarantee you at least one boy, then there aren't
> really 3 remaining possibilities. The unguaranteed
> can be a boy, or it can be a girl. What is the
> third?
In the Monty Hall case you choose a door, and it has probability 1/3 of being right. It remains 1/3 probable after he chooses a door. The other two doors together have probability 2/3 of concealing the car. It remains 2/3 after he has chosen and opened a door. The probabilities of the branches don't change in any of these cases.
Before you are told anything about the children we have four equally likely possibilities: BB, BG, GB, GG. After one is eliminated, why should the others still not be equi-probable?
> A: So now let's take another experiment.
> I tell you I'm going to flip two coins
> until at least one of them is heads. I
> do that. Now I ask you to bet on whether
> or not there is a tail. What do you
> think are fair odds?
> I misread A. I agree that it's 1/3 if you
> flip pairs until at least one is heads.
> B: Now I hunt among couples with two children
> until I find one that doesn't have two girls,
> and I ask: what are the odds they have two
> boys?
Because in Monty Hall, the effect of changing probability is produced not just via elimination but via process of elimination - which is produced only because sufficient dependencies exist for it: a) the guaranteed 1:2 ratio, b) Monty must eliminate a goat, and c) the goat he eliminates can't be behind your door. These dependencies don't exist in the children problem. Unless Monty's choice occurs under the constraint of avoiding the door we chose, nothing changes. Monty: "You door has a goat behind it" -- so what? Now both of the other doors are 50/50. No similar conflict exists in the other problem. No conflict at all exists in the other problem. Elimination occurs, but it doesn't allow you to do anything further. "No GG" is just one rule, and if one elimination rule isn't sufficient for Monty Hall probabilities to change, why here?
And elimination of what? The elimination not of a possibility, but of a probability? Does that statement even have rational meaning? I'm being dead serious - we can give it mathematical meaning quite easily, but does it even have meaning to begin with?
Yes, I apparently misread that one too. But again, I disagree on the issue of whether the form of the proposed analogue matches the original problem.
>> OK. Now B:
>> > B: Now I hunt among couples with two children
>> > until I find one that doesn't have two girls,
>> > and I ask: what are the odds they have two
>> > boys?
>> Do you agree that it's 1/3?
> Yes, I apparently misread that one too.
>> > C: I hunt among couples with two children until
>> > I find one that has at least one boy, and I
>> > ask: what are the odds they have two boys?
Let's assume for fun that you do agree with that. The original problem was:
A: I met a man who said that he has two children,
at least one of which is a boy. What is the
probability that both children are boys?
B: Among all the couples who have exactly two
children, and such that at least one of their
children is a boy, what is the probability
that both are boys?
> But again, I disagree on the issue of whether
> the form of the proposed analogue matches the
> original problem.
I've made it very clear exactly what my model is for problem B. If you claim that problem A is different, then you need to provide me with a very clear model for your interpretation.
Again, thanks. I apologize.
I'm pleased we got to the heart of it. I remember many times being in a similar situation as yours and I owe much to those too numerous to mention here who were patient with me. All I would say now is that in the future when you find yourself in disagreement, ask if you can learn something, don't just assume you are right. I found that easy, because I had more determination than talent, and was wrong often enough to be reminded constantly that there was still much to learn.
Posturing does not make your arguments more convicing.
> In GB and BG, he has only one way of telling you which is the boy. In BB, he has two. Thus, four possibilities a) GB, b) BG, c) BB and he gave you the first, d) BB and he gave you the second.
How is "ways of telling" you possibly relevant to the puzzle?
You're still apparently committing the same basic fallacy of ennumerating possibilities without assigning them probabilities and basing your argument on the wrong assumption that they must be equally probable.
> Your brain will of course refuse to process the above due to your earlier remark and the sweet, sweet motherfucking irony it would mean.
That describes exactly your behaviour in every single comment here.
With respect to only the sentence above, why is the following table incorrect?
BG 1/3 GB 1/3 BB 1/6 BB 1/6
As far as I can tell this is an accurate simulation of the problem. We have two things, coins or children, which have a 50:50 chance of being one value or another. We are told one combination of values isn't the case, so we discount them from the simulation. We then use the frequency of the target combination to approximate its probability. We haven't added any artificial ordering into the problem.
I know it's not well argued logical reasoning, but it supports the articles approach. Can you point out where I tripped up in the simulation?
[1] Of course I didn't actually sit there flipping coins. I wrote a program to do it. (let [sample-size 10000000] (/ (get (frequencies (take sample-size (filter #(not= % [:girl :girl]) (repeatedly #(vector (rand-nth [:boy :girl]) (rand-nth [:boy :girl])))))) [:boy :boy]) sample-size))
I just remembered one of them is tails, so is there is a 1/2 chance that they are both tails now? Nope it's 1/3 chance they are both tails. Either HT, TH, or TT.
By using a simulation with repetition it's easier to pose the states as 'TT' and 'not TT' and use frequencies to show 'not TT' occurs more frequently than 'TT'. I find this starts to help convince people that the states shouldn't be collapsed, even if it doesn't explicitly explain why.
Yes you have! To run the experiment, you have to treat TH and HT as different but you "pretend" not to do so by adding them into the same pile. You're pretending to know that the father tells you which child is the "known boy", but by the statement of the problem, you don't know that. Why is this an error? Because the father must follow a rule - he must give you a boy, and there are two ways for him to do that in BB but you only recognize one. If you go the "table-minus" route, which both the author and your experiment do. But you can't do that without affecting the odds! By telling you they aren't both girls, he lowers the chance that either of them might be. Yes, I'm serious.
Correct formulation of table:
a) BG; b) GB; c) BB and he gives you the first; d) BB and he gives you the second.
BG, giving the first - 1/3
GB, giving the second - 1/3
BB, giving the first - 1/6
BB, giving the second - 1/6