An off-topic question about probabilities

[sheepstache]

Although ambiguity of language has been raised as an explanation of why the answer to the  Monty Hall thing is counterintuitive I assert that even when the question is expressed unambiguously there will still be a strong tendency to latch onto the idea that the gender of the remaining child is unrelated to the  gender of the known maile child. A kind of inbuilt but faulty Occam's Razor

Yes! Thank you.

[Mississippi Mudstache]

Suppose you meet sol on the street. What is the probability that sol is male?

Right, but if you didn't know anything about me at all, the probability would be nominally 50%.  How many siblings I have, or what their genders are, would not change that number in any way.  I could tell you I had ten sisters or ten brothers and you still wouldn't know anything beyond 50/50 about my gender.

The 1/3 answer above is solely due to rephrasing the question in a way that limits the possible answers.  Don't be fooled into thinking there is any sort of predictive power about future genders of future children based on the genders of your past children.  I think some people will read this and incorrectly think "well my first kid was a boy, so there must be a 2/3 chance that my next kid will be a girl, because MDM proved it."  No he didn't.  He just made you even worse at math than you already used to be, by confusing you with a trick question.  Your next kid is still 50/50 on being a boy.

Yes, of course this is a bit of a trick question, but carefully thinking through the process of coming to the correct answer should improve, not impair, your critical thinking skills.

Let's be clear about one thing: Of course the gender of the first sibling has absolutely no impact upon the gender of the second!. That is specifically why, in the second question, when you know the gender of the oldest sibling, the probability that the second is a boy is 50%. (Let's please ignore the pedantic sidenote about how the child is obviously either one or the other. I'm surprised no one has brought up the possibility that it is a hermaphrodite...)

Of more interest is the reason why the probability of the second sibling being a boy drops to 33.3% when you don't know if it is the elder or the younger. The question could easily be reworded, as "A woman has two children. It is not the case that both children are girls. What is the probability that that both children are boys?" I think that this wording makes the answer more intuitive, even though the meaning is precisely the same! Would you agree?

[Cathy]

Yes, of course this is a bit of a trick question, but carefully thinking through the process of coming to the correct answer should improve, not impair, your critical thinking skills.

And yet just a couple posts earlier you decried critical thinking skills in favour of the reader making a laundry list of unspecified and unjustified assumptions so that the answer would work out as you wanted.

The trouble with these "trick questions" is that the people who state them rarely include all of the assumptions necessary to yield the popularised answer. For example, in the Monty Hall problem, the problem statement usually describes a particular appearance on a game show without stating that the host follows the same algorithm every time, which means that (depending on the algorithm the host uses), there may actually be no advantage to switching doors, contrary to the popularised answer.

Similarly, in your question, you require us to read "a woman has two children" as if it were equivalent to "a woman is chosen at random from among all woman with at least two children, at least one of which is a boy, and each woman satisfying those criteria has an equal chance of being chosen", and you require us to make a lengthy list of assumptions about human procreation and psychology, such as (but not limited to) assuming that whether a family will have more children is independent of the gender of past children. Those assumptions are not natural readings of what you wrote, and the only reason anybody would argue they are natural is that this is a popularised question.

I agree that analysing your question can improve critical thinking skills, but that analysis won't lead to any particular answer. It will instead lead to reflection on the nature of probability and the importance of stating assumptions precisely. It may also draw our attention to the impressive complexity of human society and the amount we need to simplify to answers questions like yours. Critical thinking is very valuable, but it doesn't have much value to if you refuse to engage with critical analysis of what you have written.

[Mississippi Mudstache]

Yes, of course this is a bit of a trick question, but carefully thinking through the process of coming to the correct answer should improve, not impair, your critical thinking skills.

And yet just a couple posts earlier you decried critical thinking skills in favour of the reader making a laundry list of unspecified and unjustified assumptions so that the answer would work out as you wanted.

The trouble with these "trick questions" is that the people who state them rarely include all of the assumptions necessary to yield the popularised answer. For example, in the Monty Hall problem, the problem statement usually describes a particular appearance on a game show without stating that the host follows the same algorithm every time, which means that (depending on the algorithm the host uses), there may actually be no advantage to switching doors, contrary to the popularised answer.

Similarly, in your question, you require us to read "a woman has two children" as if it were equivalent to "a woman is chosen at random from among all woman with at least two children, at least one of which is a boy, and each woman has an equal chance of being chosen", and you require us to make a lengthy list of assumptions about human procreation and psychology, such as (but not limited to) assuming that whether a family will have more children is independent of the gender of past children. Those assumptions are not natural readings of what you wrote, and the only reason anybody would argue they are natural is that this is a popularised question.

I agree that analysing your question can improve critical thinking skills, but that analysis won't lead to any particular answer. It will instead lead to reflection on the nature of probability and the importance of stating assumptions precisely. It may also draw our attention to the impressive complexity of human society and the amount we need to simplify to answers questions like yours. Critical thinking is very valuable, but it doesn't have much value to if you refuse to engage with critical analysis of what you have written.

Wow, I bet you're a fun dinner date.

[sol]

So if it is only a 50/50 outcome then you would be more than willing to automatically bet on the side of the prior 10 flips and happily allowme to take the other side of the bet.....right. Afterall it is even up bet in your mind so you shouldn't care what side you are on.

Yes, I would happily take that bet if the coin was fair.  Do you seriously believe that the coin has some kind of memory of its past flips that influences future flips?

Of course, a Brooklyn cabbie would rapidly surmise he was being swindled and the coin was rigged, after ten heads in a row.  But on a fair coin, past flips make no difference at all.  It could be 100 or 1000 heads in a row and the next flip is still 50/50.

Wow, I bet you're a fun dinner date.

Of all the people in this thread, I think Cathy would be the most entertaining dinner date.  I appreciate well spoken people with clearly thought out ideas much more than people who can only talk about new tv shows and sports.

[Mississippi Mudstache]

So if it is only a 50/50 outcome then you would be more than willing to automatically bet on the side of the prior 10 flips and happily allowme to take the other side of the bet.....right. Afterall it is even up bet in your mind so you shouldn't care what side you are on.

Yes, I would happily take that bet if the coin was fair.  Do you seriously believe that the coin has some kind of memory of its past flips that influences future flips?

Of course, a Brooklyn cabbie would rapidly surmise he was being swindled and the coin was rigged, after ten heads in a row.  But on a fair coin, past flips make no difference at all.  It could be 100 or 1000 heads in a row and the next flip is still 50/50.

Wow, I bet you're a fun dinner date.

Of all the people in this thread, I think Cathy would be the most entertaining dinner date.  I appreciate well spoken people with clearly thought out ideas much more than people who can only talk about new tv shows and sports.

No, you're right, that was an unfair swipe. Allow me to levy a more articulate criticism:

In this problem, you are specifically given a limited amount of information. You must calculate the probabilities of each circumstance using the information you're given. YOU DO NOT NEED MORE INFORMATION TO BE CAPABLE OF CALCULATING THE PROBABILITIES.

I am a business analyst. I love more data. It allows me to make more precise models and predictions of all sorts of useful metrics. But I don't have every single piece of data necessary to build those models. In fact, the majority of the relevant data is unknowable. Does that mean I need to throw my hands up in the air and declare every problem to which I'm tasked unsolvable? Of course not! You must make assumptions about the unknown and base your models on the known.

Since the example of playing cards was brought up earlier, let us examine a simple example. What is the probability that the top card in a standard deck of playing cards is the Ace of Spades? If this is the only information that you're given, then you must make implicit assumptions. A "standard" deck among native English speakers implies four suits (spades, hearts, clubs, and diamonds) of 13 cards each (A,2,3,4,5,6,7,8,9,J,Q,K). Therefore the answer is 1/52. Obviously, the top card is either the Ace of Spades or not the Ace of Spades, but that does not preclude us from calculating probabilities based on the information that is available to us. If it did, statistics would be a useless science. We can add more information on top of that. For example, we can state that the deck is sorted by suit, with each suit organized Ace through King. This would eliminate the possibility that 2 through Queen would be at the top of the deck. Therefore, we can refine the probability to 1/8, since we still know neither which suit is atop the deck, nor whether the King or the Ace would be atop the deck.

Now, suppose that I asked you the same question, but declared the answer to be 0%, since the "standard deck of playing cards" to which I was referring was in fact a deck of Swiss playing cards, which includes no Spades. Well, that would be a true "trick question" and ripe for criticism. But that is simply not the case in the questions that are under discussion at the moment.

I would encourage everyone to read the relevant Wikipedia article for a comprehensive discussion of the problem.

[RetiredAt63]

This is basic genetics (re sex ratio) and basic permutations and combinations theory.
Key words: Punnet square, permutations, combinations, independent variable.

Warning - you hit a hot spot, this will be longish.  Shades of my second stats lecture, way back when.

Boy/girl baby thing.  Technically the odds are not 50:50 (more boys, usually 52:48 but it does vary) in mammals (humans are mammals) in a Population.  In an individual, things differ - there are cases where a fetus is rejected (most obvious is Rh incompatibility).  But go with simplicity, 50:50, odds for one baby are 50% = 0.5 either sex.  Two independent occurrences (i.e. the two women), probability is multiplied, so 0.5x0.5 = 0.25, or 1/4.  There is only one way both babies can be girls, so .25.  There is only one way both babies can be boys, .25. But there are two ways to get one of each, either Mom A has the boy or Mom B has the boy, so that is .25 + .25 = .50.  A Punnett square shows this easily.  Mendel did the same for peas.

All this assumes the occurrences are independent.  What if you have two genes very close together on the same chromosome?  Then they will usually occur together and they are not "independently assorted".  If one gene has an abnormal form that we want to know about, the other gene may be used as a marker.

The math is easier to learn with non-living situations, because then you don't get into all the biology that complicates things.  Back to dice.
One die has 6 possibilities, right?  1, 2, 3, 4, 5, 6.  So the odds of a 1 are 1/6.  The odds of any one number are 1/6.  The odds of an even number are 1/2 (2, 4, or 6, = 1/6 + 1/6 + 1/6).  Same for the odds for an odd number.  Two dice - look at odds for each die, then add the two together for two dice.  Want a 7?  You can have 1+6, 2+5, 3+4, 4+3, 5+2, and 6 + 1.  This is easier to visualize if the two dice are two different colours, then you can see if the 1 was a white or red die.  We have 6 ways of getting a 7.  There are 36 possible combinations, so the odds are 6/36 or 1/6.  If you want a 2?  There is only one way to get a 2 = 1 + 1.  So the odds are 1/36.  For an 11, the odds are 2/36 (5+6, 6+5).

And yes, this branch of math started because gamblers wanted to know the odds.
 

[Cathy]

In this problem, you are specifically given a limited amount of information. You must calculate the probabilities of each circumstance using the information you're given. YOU DO NOT NEED MORE INFORMATION TO BE CAPABLE OF CALCULATING THE PROBABILITIES.

I think you are overlooking the difference between (i) a pure math problem and (ii) a model designed to approximate the answer to a practical problem. You are then overlooking the fact that in modelling, there isn't a necessarily single right answer since various models might be possible.

Let's say that you run into a woman on the street and she tells you that she has exactly two children, at least one of which is a boy. She then pulls out a firearm and tells you that you are going to guess about whether she has two boys and if you guess wrong, she will kill you. There is no way out of this scenario, so you have to give an answer to have any chance of living. In this case, you would look at everything you know about the woman (and perhaps about criminals in general) and try to come up with a model that would predict whether she has two boys. In such a dire situation, would be justified in crafting a series of assumptions in order to allow yourself to come up with some answer. The right answer is that she either has two boys or not, but you don't know which it is, so you come up with a model and do the best you can to maximise the chance that you walk away from this scenario alive.

The above scenario is what you say you do in your job (presumably without the threat of violence), and that makes a lot of sense. In other words, you are given a practical problem and you make some assumptions to come up with an approximate answer to it.

However, the above scenario is much different from a pure math problem. In a pure math problem, you don't introduce a bunch of assumptions that aren't stated in the problem. That's called fallacious reasoning. Given that this thread is stated to be a "question about probabilities", it was reasonable to expect that this was an exchange of pure math problems, not "please come up with any set of assumptions not stated in the problem to derive any answer".

You may say that your problem statement obviously intended for the reader to come up with a ton of assumptions in order to get the answer you wanted, but I disagree. Most importantly, there is no single right model for your problem. In the life or death scenario above, different people would use different models. One person might approach it like you want. Another person, however, might be a census researcher and might be aware that the neighbourhood where this is taking place has an unusually high concentratration of families with two boys; and the researcher might further speculate that the criminal is also from this neighbourhood, perhaps due to a logo on a shirt she is wearing. Based on that, the answer might be totally different from the one you want.

I'm not really sure how to explain this any clearer. You seem to be under the impression that your question necessarily leads to a natural set of assumptions that in turn yield the popularised answer. However, that is just not the case. Arriving at your intended answer requires an extremely specific set of assumptions. In an abstract sense, it's just as valid to come up with other assumptions instead.

[Mississippi Mudstache]

You may say that your problem statement obviously intended for the reader to come up with a ton of assumptions in order to get the answer you wanted, but I disagree. Most importantly, there is no single right model for your problem. In the life or death scenario above, different people would use different models. One person might approach it like you want. Another person, however, might be a census researcher and might be aware that the neighbourhood where this is taking place has an unusually high concentratration of families with two boys; and the researcher might further speculate that the criminal is also from this neighbourhood, perhaps due to a logo on a shirt she is wearing. Based on that, the answer might be totally different from the one you want.

I'm not really sure how to explain this any clearer. You seem to be under the impression that your question necessarily leads to a natural set of assumptions that in turn yield the popularised answer. However, that is just not the case. Arriving at your intended answer requires an extremely specific set of assumptions. In an abstract sense, it's just as valid to come up with other assumptions instead.

No, no, no. The problem does not state which specific neighborhood the person is from or what t-shirt they are wearing. If you were supposed to take that information into account, then I would have stated that information! If the woman in question was a specific person, and not a person chosen at random who happened to have two children, then I would have specified who the person was as well. I don't know how to explain it any clearer, either!

Without making completely ridiculous assumptions, there is no possible way that your answers can different significantly from 1) 33.3% and 2) 50%. It makes no material difference if the biological birth rate for mammals is 52% males and 48% females, or if 1% of the population is hermaphroditic. Trust me, I understand the point you are trying to make, but you're are missing the forest for the trees (or at least you're pretending to in order to make your point).

To be clear, there is no laundry list of assumptions that must be made. The assumptions are:

• The probability that a randomly selected child is male is 50/50.
• The woman in question has been randomly selected from the population of mothers who have two children.

No other assumptions are necessary.

[Cathy]

To be clear, there is no laundry list of assumptions that must be made. The assumptions are:

• The probability that a randomly selected child is male is 50/50.
• The woman in question has been randomly selected from the population of mothers who have two children.

No other assumptions are necessary.

This is wrong, which kind of illustrates my point. Many more assumptions are necessary to yield the popularised answer. Earlier in the thread I already gave an example of how the answer can be made to be arbitrarily close to 0 or 1, but still consistent with both premises you mention. In fact, a trivial variation of my earlier post allows the answer to be made to be any value. We could even make a follow-up question out of that, similar to the one I posted regarding the Monty Hall problem, although I suspect no one other than MDM would attempt it.


Mississippi Mudstache, the core claim you make is that your question leads to a natural set of assumptions and anything else is "ridiculous". I understand your claim, but I disagree with it for the reasons stated in my various past posts.

[Mississippi Mudstache]

To be clear, there is no laundry list of assumptions that must be made. The assumptions are:

• The probability that a randomly selected child is male is 50/50.
• The woman in question has been randomly selected from the population of mothers who have two children.

No other assumptions are necessary.

This is wrong, which kind of illustrates my point. Many more assumptions are necessary to yield the popularised answer. Earlier in the thread I already gave an example of how the answer can be made to be arbitrarily close to 0 or 1, but still consistent with both premises you mention.

I purposefully ignored that example, because I do indeed find it ridiculous. I'm okay with you disagreeing, because I'm at least convinced that we fully understand one another's arguments at this point.

[TheOldestYoungMan]

Wow, I bet you're a fun dinner date.

I cannot imagine a better night out then discussing something with Cathy.

[TheOldestYoungMan]

I would encourage everyone to read the relevant Wikipedia article for a comprehensive discussion of the problem.

Well...right...from the wikipedia article:

In response to reader criticism of the question posed in 1959, Gardner agreed that a precise formulation of the question is critical to getting different answers for question 1 and 2. Specifically, Gardner argued that a "failure to specify the randomizing procedure" could lead readers to interpret the question in two distinct ways:

From all families with two children, at least one of whom is a boy, a family is chosen at random. This would yield the answer of 1/3.
From all families with two children, one child is selected at random, and the sex of that child is specified to be a boy. This would yield an answer of 1/2.

...

The paradox occurs when it is not known how the statement "at least one is a boy" was generated. Either answer could be correct, based on what is assumed.[12] However, the "1/3" answer is obtained only by assuming P(ALOB|BG)=P(ALOB|GB)=1, which implies P(ALOG|BG)=P(ALOG|GB)=0. As Marks and Smith say, "This extreme assumption is never included in the presentation of the two-child problem, however, and is surely not what people have in mind when they present it."

So what about posing a generally acknowledged ambiguous question is a defense of it being unambiguous...

The point of discussing this particular problem in probability and statistics is because of the ambiguity.  It's to drive home to those studying the importance of stating any and all assumptions, both when you collect the data that will be used in statistical analysis, and when you perform the analysis.

More from the article:

From the position of statistical analysis the relevant question is often ambiguous and as such there is no “correct” answer.

Fox & Levav (2004) used the problem (called the Mr. Smith problem, credited to Gardner, but not worded exactly the same as Gardner's version) to test theories of how people estimate conditional probabilities.[2] In this study, the paradox was posed to participants in two ways:

"Mr. Smith says: 'I have two children and at least one of them is a boy.' Given this information, what is the probability that the other child is a boy?"
"Mr. Smith says: 'I have two children and it is not the case that they are both girls.' Given this information, what is the probability that both children are boys?"
The authors argue that the first formulation gives the reader the mistaken impression that there are two possible outcomes for the "other child",[2] whereas the second formulation gives the reader the impression that there are four possible outcomes, of which one has been rejected (resulting in 1/3 being the probability of both children being boys, as there are 3 remaining possible outcomes, only one of which is that both of the children are boys). The study found that 85% of participants answered 1/2 for the first formulation, while only 39% responded that way to the second formulation. The authors argued that the reason people respond differently to each question (along with other similar problems, such as the Monty Hall Problem and the Bertrand's box paradox) is because of the use of naive heuristics that fail to properly define the number of possible outcomes.

If you do define your assumptions, then your conclusion could be deemed correct or incorrect, depending on an independent evaluation of the conclusion's self-consistency at that point.

[RidinTheAsama]

To be clear, there is no laundry list of assumptions that must be made. The assumptions are:

• The probability that a randomly selected child is male is 50/50.
• The woman in question has been randomly selected from the population of mothers who have two children.

No other assumptions are necessary.

This is wrong, which kind of illustrates my point. Many more assumptions are necessary to yield the popularised answer. Earlier in the thread I already gave an example of how the answer can be made to be arbitrarily close to 0 or 1, but still consistent with both premises you mention. In fact, a trivial variation of my earlier post allows the answer to be made to be any value. We could even make a follow-up question out of that, similar to the one I posted regarding the Monty Hall problem, although I suspect no one other than MDM would attempt it.


Mississippi Mudstache, the core claim you make is that your question leads to a natural set of assumptions and anything else is "ridiculous". I understand your claim, but I disagree with it for the reasons stated in my various past posts.

I've been reading along and (I think) I can see both of your sides to this completely clearly... I'm so interested in what conditions or statements could finally have you both saying 'yes I agree'.

Cathy,
what would be your take on the question if the statement "no other information is available" had been added, thereby ruling out all potential unknown census and neighbourhood data?

It seems that the key sticking point between you two is the level of assumptions that must be listed before this can be considered a legitimate mathematical question.

So again, to Cathy,
Where would you draw the line?  Clearly you are well versed on the topic and it sounds as if you review issues that have mathematical similarities at least on occasion.  So in those instances, how long is the list of assumptions that you require?  Do you regularly state "It is assumed that this problem takes place on Earth", "It is assumed that Superman does not exist", "It is assumed that gravity will not reverse during the course of the event", etc. etc. ?  In my opinion, the assumption statement "It is assumed that no further information regarding the mother's life-situation is or ever will be available" would have been totally unnecessary in the original question, but you point to other such information as a major factor in why the question is not answerable...

I suppose my understanding of this more readily aligns with that of Mississippi Mudstache, which is why my questions go to Cathy.  But I am truly interested in finding the middle ground between your viewpoints.  If it exists...

[brooklynguy]

Clearly you are well versed on the topic

Spend enough time here and you will soon learn there is not one among the broad array of diverse topics in which Cathy chooses to participate about which this statement would not hold true (which is to say, count me among those who believe she would make an interesting dinner date).

[Mississippi Mudstache]

Well...right...from the wikipedia article:

In response to reader criticism of the question posed in 1959, Gardner agreed that a precise formulation of the question is critical to getting different answers for question 1 and 2. Specifically, Gardner argued that a "failure to specify the randomizing procedure" could lead readers to interpret the question in two distinct ways:

From all families with two children, at least one of whom is a boy, a family is chosen at random. This would yield the answer of 1/3.
From all families with two children, one child is selected at random, and the sex of that child is specified to be a boy. This would yield an answer of 1/2.

Okay, this is actually a very good point, and one that has not been specifically addressed yet. Which means that my prior statement about the necessary assumptions was flawed:

To be clear, there is no laundry list of assumptions that must be made. The assumptions are:

• The probability that a randomly selected child is male is 50/50.
• The woman in question has been randomly selected from the population of mothers who have two children.

The second condition must therefore be appended with the phrase "at least one of which is a boy".

As noted later in the article, "A survey such as vos Savant's suggests that the majority of people adopt an understanding of Gardner's problem that if they were consistent would lead them to the 1/3 probability answer but overwhelmingly people intuitively arrive at the 1/2 probability answer. Ambiguity notwithstanding, this makes the problem of interest to psychological researchers who seek to understand how humans estimate probability." The survey referenced is vos Savant's query to readers who had two children, at least one of whom was a boy. 35.9% of readers (~1/3) reported two boys.

So even though the majority of people understand the implied assumptions correctly, the majority, even working with the correct assumptions, arrive at the wrong answer. That's why this is an interesting problem.

[Cathy]

Okay, this is actually a very good point, and one that has not been specifically addressed yet.

This was addressed in my first post in the thread and then again in a post that you quoted and criticised and then again in various follow-up posts.

Your question, as stated by you earlier in this thread (and even in the introduction to the Wikipedia article), does not explicitly involve any random variable at all. It just calls our attention to a particular woman with already-existing children. This only becomes a probability question if we assume that when you say "a woman has" you really mean to introduce a variable random, but the distribution of that random variable was never defined. It's actually possible to make the answer be any value by choosing the distribution of the unspecified random variable accordingly. Alternatively, if the question is taken literally and not read to introduce an unspecified random variable, the answer is 0 or 1.

Cathy,
what would be your take on the question if the statement "no other information is available" had been added, thereby ruling out all potential unknown census and neighbourhood data?

If "no other information is available", it's not a probability question because there is no random variable. This only becomes a probability question if we assume that when you say "a woman has" you really mean to introduce a variable random, but the distribution of that random variable was never defined. As I previously explained in the thread, in certain contexts, a random variable can be implicitly introduced in passing when there is only one possible interpretation. For example, to use one of my previous examples, in the context of card games, it is often clear what random variable is being discussed; but it is certainly not clear in the gender question what random variable (if any) is involved, which is the one reason it is ambiguous (although not the only reason).

[RidinTheAsama]

Okay, this is actually a very good point, and one that has not been specifically addressed yet.

This was addressed in my first post in the thread and then again in a post that you quoted and criticised and then again in various follow-up posts.

Your question, as stated by you earlier in this thread (and even in the introduction to the Wikipedia article), does not explicitly involve any random variable at all. It just calls our attention to a particular woman with already-existing children. This only becomes a probability question if we assume that when you say "a woman has" you really mean to introduce a variable random, but the distribution of that random variable was never defined. It's actually possible to make the answer be any value by choosing the distribution of the unspecified random variable accordingly. Alternatively, if the question is taken literally and not read to introduce an unspecified random variable, the answer is 0 or 1.

Cathy,
what would be your take on the question if the statement "no other information is available" had been added, thereby ruling out all potential unknown census and neighbourhood data?

If "no other information is available", it's not a probability question because there is no random variable. This only becomes a probability question if we assume that when you say "a woman has" you really mean to introduce a variable random, but the distribution of that random variable was never defined. As I previously explained in the thread, in certain contexts, a random variable can be implicitly introduced in passing when there is only one possible interpretation. For example, to use one of my previous examples, in the context of card games, it is often clear what random variable is being discussed; but it is certainly not clear in the gender question what random variable (if any) is involved, which is the one reason it is ambiguous (although not the only reason).

I'm getting closer to understanding your points...
Could you provide an example of how you would phrase the question (a woman who has two children, at least one of which is a boy, and we are wondering about the chances of both children being boys) in a succinct manner that does properly introduce a random variable?  I think seeing the 'same question' written in a way that you consider to be correct would help very much.  Thanks in advance!

[Cathy]

Could you provide an example of how you would phrase the question ... in a succinct manner...

It's easy to phrase this question in an unambiguous way.

One way to do it would be: "Consider a family with two children. A priori, there is a 1/4 chance that both children are male, a 1/4 chance that both children are female, and a 1/2 chance that there is one child of each gender. Now further suppose that at least one of this family's children is male. In light of this supposition, what is the chance that both children are male?"

If stated that way, the answer is calculated by (1/4) / (1/4+1/2) = 1/3, which is the popularised answer. You will notice that my statement is short and addresses every issue I have raised throughout this thread. The reason you never see the question phrased in this unambiguous manner is that then the answer would be obvious, and the purpose of the question is supposed to be to trick people, not to convey unambiguous meaning.

[RidinTheAsama]

Could you provide an example of how you would phrase the question ... in a succinct manner...

It's easy to phrase this question in an unambiguous way.

One way to do it would be: "Consider a family with two children. A priori, there is a 1/4 chance that both children are male, a 1/4 chance that both children are female, and a 1/2 chance that there is one child of each gender. Now further suppose that at least one of this family's children is male. In light of this supposition, what is the chance that both children are male?"

If stated that way, the answer is calculated by (1/4) / (1/4+1/2) = 1/3, which is the popularised answer. You will notice that my statement is short and addresses every issue I have raised throughout this thread. The reason you never see the question phrased in this unambiguous manner is that then the answer would be obvious, and the purpose of the question is supposed to be to trick people, not to convey unambiguous meaning.

I would say we don't see it phrased this way, not because it lacks trickery, but because it is boring. (No offence intended).
As I see the original question, the intent is not to trick people, but to force them to think through more than one step in solving a problem.  Why much each possible scenario be explained, along with its corresponding probability, in order to be an unambiguous question? 

Your version of the question takes the assumption that any child has a 50/50 chance of being born male or female, and the full list of potential birth patterns - and "pre-solves" this portion for the one to whom the question gets posed, leaving them only with the basic math to deal with. Without a doubt this makes it unambiguous, but also makes it much more simple.  Is there not a way to remain unambiguous, while also leaving a multi-step thinking process for the person solving the question? (ie. step 1 - determine full list of potential scenarios and probability of each, step 2 - based on given information that rules out certain scenarios, calculate new probability of target scenario). 

[Cathy]

Your version of the question takes the assumption that any child has a 50/50 chance of being born male or female, and the full list of potential birth patterns - and "pre-solves" this portion for the one to whom the question gets posed, leaving them only with the basic math to deal with. Without a doubt this makes it unambiguous, but also makes it much more simple.  Is there not a way to remain unambiguous, while also leaving a multi-step thinking process for the person solving the question? (ie. step 1 - determine full list of potential scenarios and probability of each, step 2 - based on given information that rules out certain scenarios, calculate new probability of target scenario).

The reason I spelled out the probability of each of the patterns is that you cannot derive that information from the fact that any given child has a 50% chance of being male. Even if we accept that any given child has a 50% chance of being male, it's still possible that, say, 80% of families with exactly two children might have two male children. I gave one example of how this sort of thing can arise, but you can use your imagination to come up with others. This is one of the core reasons that the original problem is not well-formulated, and it's intentionally fixed in my version.

It is possible to fix that flaw without explicitly defining the probability of each pattern, but it would then require quite a few premises and it is very tricky to come up with a list of premises that both works and is fairly minimal. I don't feel like doing that, so I'll leave it as an exercise.

[RidinTheAsama]

Your version of the question takes the assumption that any child has a 50/50 chance of being born male or female, and the full list of potential birth patterns - and "pre-solves" this portion for the one to whom the question gets posed, leaving them only with the basic math to deal with. Without a doubt this makes it unambiguous, but also makes it much more simple.  Is there not a way to remain unambiguous, while also leaving a multi-step thinking process for the person solving the question? (ie. step 1 - determine full list of potential scenarios and probability of each, step 2 - based on given information that rules out certain scenarios, calculate new probability of target scenario).

The reason I spelled out the probability of each of the patterns is that you cannot derive that information from the fact that any given child has a 50% chance of being male. Even if we accept that any given child has a 50% chance of being male, it's still possible that, say, 80% of families with exactly two children might have two male children. I gave one example of how this sort of thing can arise, but you can use your imagination to come up with others. This is one of the core reasons that the original problem is not well-formulated, and it's intentionally fixed in my version.

It is possible to fix that flaw without explicitly defining the probability of each pattern, but it would then require quite a few premises and it is very tricky to come up with a list of premises that both works and is fairly minimal. I don't feel like doing that, so I'll leave it as an exercise.

I thought my previous suggestion of a statement along the lines of "no additional information [about family demographics] is available" would be sufficient for this.  Does this not rule out all potential scenarios where families strive for specific family patterns?  Where mortality rates are different for male and female children?  Where Aliens abduct the first born male in every household? etc?

Would the following meet your criteria for a properly phrased question?

Suppose you meet a mother who tells you she has exactly 2 children, and that at least one of them is a boy.  With the assumptions that children have a 50% chance of being born male, and that the genders of all suchmother's children are not influenced by any other factors, what is the probability that you have met a mother of 2 boys?

Have I properly introduced a random variable?
Have I sufficiently ruled out all "oddball" scenarios where MM, MF, FM, FF are not equally probably to each other?

[Cathy]

Your new formulation has some of the same flaws that I already criticised before (for example, you don't tell us how we came across this "mother" which, as before, means that the distribution of the random variable is unspecified; my formulation avoided that issue and other issues). Your new formulation also has some novel flaws. However, overall, I don't think there's anything new to add at this point. I've shown you an unambiguous way to state the problem. You can review my previous posts for all the answers to your various questions.

[TheOldestYoungMan]

You end up in a "reasonable assumptions" wasteland.  In the course of a full length study of the subject, an instructor might get into the habit of making the type of assumptions you think follow from a statement like "no other information is available."  Which leads some students to think that those assumptions are somehow more reasonable than others, and indeed this sentiment may be inadvertently reinforced by the instructor.  In reality the assumptions in classwork are assumed for convenience, not for any righteous reasons.  The only reasonable assumptions are those backed up by data.

So when you study this problem, you make assumptions to make the math easier to isolate.

The best example I can think of is physics, where a common assumption is a friction-less environment.  This is made for virtually all physics problems where you want to simplify the concept to ignore friction.  It is not in any way a reasonable assumption, and neglecting it is only useful if you are content to get the wrong answers, while you verify you understand how one part of the universe works with easier math.  It gets referred to as a reasonable assumption, but it isn't.  You can't have a friction-less environment.  You neglect the term because carrying it through all of your equations is tedious, and you haven't yet learned how to incorporate it into the functions.  It is no more valid to assume no friction than it is to assume no mass or to assume penguins have thumbs.  But once an assumption is made, a solution can be evaluated to see if it is self-consistently correct.

But even when you do this in physics, for purely classroom convenience, the disciplined instructor will require you to explicitly state that assumption for each and every problem (where not included in the problem statement). 

It is with much fear and trepidation that the advanced engineering student sees the list of allowable "reasonable" assumptions drop off one by one, until they are left in the cold and desolate wasteland of having to account for everything with hard data.

I imagine it is much the same for those that study statistical analysis beyond the elementary discussions of when to add and when to multiply.

[RidinTheAsama]

Your new formulation has some of the same flaws that I already criticised before (for example, you don't tell us how we came across this "mother"

In the absence of a description of how the meeting came about along with the statement that no such information is available, all possible modes of meeting that would lead you to expect a higher proportion of males are equally probable to modes of meeting that would lead you to expect higher proportion of females.  So I see no need to qualify how the meeting was initiated.

Your new formulation also has some novel flaws.

Like what?

I've shown you an unambiguous way to state the problem.

No, you've shown us an unambiguous way to state HALF of the problem.  You left out a way for the questioner to push the questionee to think for themselves about what scenarios are possible, and what scenarios should be ruled out.

However, overall, I don't think there's anything new to add at this point. ...... You can review my previous posts for all the answers to your various questions.

I understand if you have become bored of the topic or fed up with me specifically... And I apologize if it's the latter.
If you choose to add no more, that's fair.

But if you or anyone else is still willing, I'm still interested in finding the balance between the ambiguous but interesting multi-step problem that leaves no correct answer, and the unambiguous but striped-of-thought-provocation version.

[EscapeVelocity2020]

(in honor of Arebelspy)

[Playing with Fire UK]

1. https://www.xkcd.com/169/

2. 11th head - consider also which is more likely, that a fair coin will come up with 10 heads in a row or that I am lying to you about it being a fair coin in order to win a bet in the pub?

3. the n options so therefore odds of each is 1/n is a well established error in people who haven't had enough stats training, and is the origin of the 33%. My personal method of explaining the error is to add an option that is clearly improbably. So in this case, if there are three gender combination possibilities (considering that M+F == F+M), and we add the option of the women giving birth to frogs, would that change the odds to 1/4? If we add the option of giving birth to a teapot does that now make the likelihood of two boys 1/5?

Now a biology question please (not asking about the stats): My Greatgrandmother, Grandmother, Mother and her sisters, and my sisters have only given birth to girls. There have been a low number of miscarriages and no identical twins, no social pressures for selective abortion. I know that the gender of a child is determined by the sperm, but what biological/medical reasons could there be to cause this? Would something like being carriers of a dominant gender specific disease cause this pattern of births? I occassionally wonder if this is a 10 heads cluster or a biological double headed coin?

[RidinTheAsama]

Now a biology question please (not asking about the stats): My Greatgrandmother, Grandmother, Mother and her sisters, and my sisters have only given birth to girls. There have been a low number of miscarriages and no identical twins, no social pressures for selective abortion. I know that the gender of a child is determined by the sperm, but what biological/medical reasons could there be to cause this? Would something like being carriers of a dominant gender specific disease cause this pattern of births? I occassionally wonder if this is a 10 heads cluster or a biological double headed coin?

I have nothing to back this up with... just talk that I've heard, so take it with a grain of salt, but....

Supposedly you can "choose" the gender of your child based on when you have sex during the ovulation cycle.  They say 'male sperms' are faster so if their race begins when the egg is already in striking distance, a male offspring will result.  On the flip side, 'female sperms' are slower but active for much longer, so if the race begins a few days before the egg is available, the 'male sperms' die off and it will be a 'female sperm' who fertilizes the egg.

Again, no idea how true that is, I've never even bothered with so much as a google...

But, if it is true, perhaps there's a trait in your family for the females to be more interested in sex at a time in their cycle slightly before ovulation occurs?