A Probability Puzzle, Part Two

The response to Tuesday’s post, currently at 97 comments, has been very interesting. Since some of the commenters appear to be growing restless, I will put off until tomorrow my epic Iraq war post (based on my having recently waded through all 482 pages of Thomas Ricks’ subtly titled book Fiasco) and talk about probability instead.

Here’s the puzzle, in case you missed it the first time:

A shopkeeper says she has two new baby beagles to show you, but she doesn’t know whether they’re both male, both female, or one of each. You tell her that you want only a male, and she telephones the fellow who’s giving them a bath. “Is at least one a male?” she asks him. She receives a reply. “Yes!” she informs you with a smile. What is the probability that the other one is a male?


Reduced to its essence, the problem is asking this: “There are two dogs. One of them is male. What is the probability that the other one is also male?”

ANSWER: The probability is 1/3.

To see this, let us call one of the dogs X and the other Y. Initially we have four equally likely possibilities:

  1. X=Male, Y=Male
  2. X=Male, Y=Female
  3. X=Female, Y=Male
  4. X=Female, Y=Female

This is our starting point. If the question were simply, “What is the probability that two randomly chosen dogs are both males?” then the correct answer would be 1/4. (This would be a good time to state explicitly that we assume that males and females are equally likely.)

Now we receive information that allows us to update our probabilities. We learn that one of the dogs is a male. That means that option four now has a probability of zero. But this information does nothing to affect our assessments that the first three possibilities are equally likely. That is, the revelation that one of the dogs is a male is true regardless of which of the three scenarios we are in.

So we have three, equally likely possibilities, and in only one of them are both dogs male. So the answer is 1/3.

We could also approach this via Bayes’ Theorem. Let A=Both dogs are male and let B=One of the dogs is male. We seek P(A|B). (That is, we seek the probability that A has occurred, given that B is known to have occurred.)

According to Bayes Theorem, P(A|B) is equal to P(B|A) times P(A) divided by P(B).

  • P(B|A) is 1. If we know both dogs are male, then it is a sure thing that one of the dogs is male.
  • P(A) is 1/4. The sexes of the dogs are determined independently of one another. Since there is a probability of 1/2 that each dog individually is male, the probability that both are male is obtained by multiplying 1/2 by itself.
  • P(B)=3/4. The only way for B not to occur is for both dogs to be female. Since that only happens 1/4 of the time, we see that B occurs 3/4 of the time.

Plugging this all in to Bayes’ Theorem gives us (1 times 1/4) divided by 3/4, which simplifies to 1/3.

A common incorrect answer is 1/2. The reasoning is that the probability that any given dog is male is 1/2, and the sexes of the two dogs are determined independently of one another. Therefore, knowing that one of the dogs is male effectively allows us to eliminate options three and four from our list. That leaves only the first two, equally likely, possibilities.

But there is a subtle, extra assumption being made in this argument. The reasoning above woud be valid if we were told specifically that dog X was male. If we ask something like, “If the dog on the chair is male, what is the probability that the dog standing by the door is also male?” then the answer is, indeed, 1/2. In that case we would be asking, in effect, “Does knowing that this particular dog over here is a male affect the probability that that dog over there is a male?” The answer is no. In this situation we are asking questions about the characteristics of individual dogs.

That is not the situation we have in this problem. In our case the question involves the characteristics of a pair of dogs. We are asking, in effect, “What is the probability that both dogs are male given that one of them is male?” The piece of information we receive from the shopkeeper does not relate to a specific, individual dog. Rather, it tells us something about the pair of dogs. By dividing the pair into two individual dogs, singling out one of them, and applying the shopkeeper’s information to that particular dog, we are making assumptions beyond what is given to us.

Another common incorrect answer is 1/4. Here the idea is that the probability that both dogs are male is “set in stone” so to speak. That is, the probability that two randomly chosen dogs are both male is 1/4, and simply being told that one of them really is male does nothing to change that fact. The error here is that knowing that one of the dogs is male means that the pair of dogs was not randomly chosen. Conclusions drawn about pairs of radnomly chosen dogs are therefore no longer applicable.

Here’s one more way of looking at it. Imagine that you flip two coins simultaneously. You do this multiple times, keeping track of the number of times you flip the coins. Every time you get two heads you make a little mark on a piece of paper. But here’s the catch: any time two tails come up you pretend like that flip never happened. You do not include it in your count of the number of times you flipped the coins. I am saying that if you do this a couple of hundred times, being careful to flip the coins fairly, then when you are done the number of marks on the page will be close to 1/3 of the number of tosses. Try it and see.

Browsing through the comments to the original post, I noticed a few challenges to the wording of the problem. One commenter suggested that the question was a bit misleading, since the term “the other one,&rdquo is vague. The idea is that to talk about “the other one” we must first single out one of the dogs, and that has not been done.

I don’t think this is correct, however. One of the dogs has been singled out, we simply don’t know which one. The person giving the dogs a bath has asserted that one of the dogs is male. If the shopkeeper asks him to prove it, he would point to one of the dogs and say, “See! That one is male.” Presumably that is what he did in answering the question in the first place. So the groomer has singled out one of the dogs by answering the question, we just don’t know which dog it was.

I would agree, however, that if you take the final question by itself, divorced from the rest of the problem, then you have a meaningless question. It makes no sense to talk about “the other one” unless you have a pair of things and one item in the pair has been singled out. I would suggest, however, that that is a rather bizarre way of reading the problem. In context I don’t think there is any ambiguity about what is intended. In the future, however, Ill take yet another commenter’s suggestion and phrase the question, “What is the probability that the other dog is also a male.”

Another commenter asked whether the dogwasher had seen more than one of the dogs. Actually, it’s irrelevant how many dogs the dogwasher saw. All that matters is that he has passed along to you the presumably correct piece of information that one of the dogs is a male.

Hopefully I’ve addressed most of the main threads that various people brought up. But if I haven’t, just let me know and perhaps we’ll do a third installment!

Comments

  1. #1 Mike
    December 29, 2006

    Your reasoning is incorrect. This is the classic Monty Hall trap. You are making assumptions about the probability distribution the bath guy is selecting his responses from. It might be that he only had time to check one of the dogs, in which case we have no information whatsoever about the “other” dog. Your answer requires the assumption that both dogs were checked and the bath guy is being “cute” with his response. Realistically, its more likely he would have answered “yes one is” or “yes both are” if he had checked both.

  2. #2 Another-Mike
    December 29, 2006

    @Mike: Your complaint makes no sense because the ‘Yes!’ statement is provided by the shopkeeper, not the dog bather. There’s a certain amount of information loss going on here – compounded by the shopkeeper choosing to respond in a Yes/No fashion (making bickering over “Yes, at least one is” vs. “Yes, both are” a moot point).

    In addition, allowing for the notion of ‘only having time to check one dog’ creates the problem of unintentional falsehoods (ie. if the shopkeeper had said ‘No!’ instead) and for the purposes of this game we’re assuming everyone is telling the actual truth.

  3. #3 Richard Wein
    December 29, 2006

    Although there is only one sensible interpretation of the question (and even more so with your revised formulation), I still maintain that the question (even in the revised formulation) does not make sense from a strictly logical (if pedantic) point of view.

    As I said in the earlier thread, “the other one” is undefined if neither one has been specified previously. “At least one” is a numeric expression, not a specification of a particular dog. Suppose that _both_ the dogs are male. Then which dog does “the other one” refer to? (Yes, in that case it doesn’t matter which one it refers to. But the question here is whether it’s meaningful, not whether it matters.)

    If you’re going to reword the question in future, you might as well avoid the problem entirely by writing “What is the probability that both dogs are male?”

  4. #4 Russell Blackford
    December 29, 2006

    Though I ended up in the p= 1/3 corner of the other thread, for reasons to similar to those Jason has now given, I do think it would be better to say “What is the probability that both dogs are male?” The other wording tends to focus readers on the idea that a particular dog is under discussion. I suspect less people would go down the “p = 1/2″ path, as I initially did (alas!) without that distraction.

  5. #5 Caledonian
    December 29, 2006

    Humans just aren’t any good at intuitively dealing with probabilities.

  6. #6 David D.G.
    December 29, 2006

    Thanks for providing your solution, Jason.

    However, I still hold by my opinion that the probability cannot be anything but 1/2. (Yeah, I know, this is not my line of expertise at all, so I’ll defer to you that yours is the “official” answer — but that doesn’t mean I agree with it.)

    For one thing, how the heck are F/M and M/F considered separate possibilities? They’re the same freakin’ thing, at least from the perspective of the person wanting to know about the sex of the dogs; the order of birth, order of checking the dogs, or whatever makes no difference — it’s completely commutative, if I am remembering the correct use of that word from elementary school math.

    When the dog bather checks the puppies, it makes no difference (since we get our information from the fellow talking to him on the phone) which one he checks first. So once those are combined into a single possibility, as they should be, then we are left with only two choices: M/M and (F/M or M/F). So the result is still a 50/50 probability, as I see it.

    ~David D.G.

  7. #7 mark
    December 29, 2006

    Okay, I’ve changed my mind–it doesn’t matter if the dogwasher has seen both or only one dog, because he has answered that at least one is male. So, as you note, the F-F combination can be eliminated for consideration of the other dog’s sex. If he only saw 1 dog and it was male, it could have been from the M-M, M-F, or F-M combination, resulting in a 1/3 chance (M-M) the other pup is male. If the dog he saw was female, his answer would not be truthful. So the answer does indeed reduce to Reduced to its essence, the problem is asking this: �There are two dogs. One of them is male. What is the probability that the other one is also male?�

  8. #8 Jeff
    December 29, 2006

    Man, I hate being stubborn but I’m in full agreement David D.G.

    I see how the 1/3 answer is justified mathematically, but I think the 1/2 is more logical because the bather would either check one dog (when first is male) or both dogs (when first is female). We’d know for sure that one dog is male, and this leaves a 50/50 chance the other is male.

    I think there’s a certain degree of ambiguity in the wording, unfortunately.

  9. #9 W. Kevin Vicklund
    December 29, 2006

    David, step back from the problem. Ignore the question for a moment.

    Good. Now, let’s set up a different problem.

    Given two dogs:

    What is the probability that none are male?

    What is the probability that exactly one is male?

    What is the probability that both are male?

    If we accept your argument, that M/F and F/M are not separate possibilities, then the answers to all three questions is 1/3.

    Do you believe that that is the correct answer? Or should it be 1/4, 1/2, 1/4? If the latter, why?

  10. #10 David Heddle
    December 29, 2006

    Nice problem!

    I loved teaching probability. One of my favorite questions was:

    China has a hated one-child policy, which is especially despised in the rural areas. Suppose, as a compromise, they adopted a one son policy: couples can keep having children until they have one son, then they must stop.

    What would this do to the distribution of boys and girls in China?

    Another one:

    Bill Buckner and Mookie Wilson had a free-throw competition. Since they live on different coasts, they just reported by phone how well they did.

    In the first round, Mookie made 40% of his shots, and Bill only made 35%
    In the second round, Mookie made 25% of his shots, and Bill only made 20%

    After the second round, Mookie declared himself the winner.

    Question: was he justified in doing so? If not, why? (You may assume total honesty in their reporting.)

  11. #11 David Holland
    December 29, 2006

    David D.G. and Jeff,
    Get your coins out and start flipping. Let us know how it works out.

  12. #12 Jeff Chamberlain
    December 29, 2006

    “Humans just aren’t any good at intuitively dealing with probabilities.” (Caledonian.)

    I think it may be worse than that. Some humans (myself included) seem to have trouble understanding probabilities (at least of the “Monty Hall problem” variety) even when mathematicians explain them.

    Is there there something peculiar about the brains of such people (assuming that they investigate the problems in good faith and attentively and with sufficient intelligence)? Or is there something lacking the the way mathematicians explain probabilities to the “laity?” Or (of course) some combination of both? Or what?

  13. #13 W. Kevin Vicklund
    December 29, 2006

    Sorry Jeff, 1/2 is more intuitive, but not more logical.

    Let’s re-examine your argument. If the bather only checks one dog (meaning the first checked dog is male), there are two possibilities – the second dog checked is either male or female. Now, if he has to check two dogs (meaning the first is female), there is only one possibility – the second dog checked is male. That gives us three possibilities, only one of which wins.

  14. #14 AJS
    December 29, 2006

    Jeff,

    We aren’t told, and must not attempt to guess, how many dogs the dog bather examined. All we know is that at least one is male. It could be that the first one examined is male, and the second one need not be examined; or it could be that the first one examined was female and the second was male.

    David,

    There are four outcomes: first-examined M, second-examined M; first-examined M, second-examined F; first-examined F, second-examined M; first-examined F, second-examined F. Thus, there are two equally-probable ways to get one male and one female. (The other two equally-probable outcomes being M-M and F-F). But we know that “both female” isn’t a permitted outcome, so the probability of both dogs being male is 1/3 and not 1/4.

    It’s crucial that we don’t know whether the confirmed-male dog is the first-examined one or the second-examined one. If we asked “Was the first dog you examined male?” and were told yes, then the probability that both dogs are male is equal to the probability that the second dog is male, = 1/2. But that wasn’t what we asked: we allowed for the case where the first dog examined could be female and the second male.

  15. #15 brtkrbzhnv
    December 29, 2006

    David D.G.: You are assuming that the MF (including FM) and MM events are equally likely. This is not the case. If you flip a pair of coins a million times you will get HT/TH about 500 000 times, but you’ll get HH only about 250 000 times. Discounting the 250 000 TTs, we thus get 1/3 HH and 2/3 HT/TH.

  16. #16 hrln
    December 29, 2006

    Yes that is a very quaint problem. However, I understand everybody’s confusion. You did not define any event set in your structuring of the problem. Technically, this question is identical in mathematics to your wording:

    two people are walking down the street. one is a man. what is the probability that the other is a man?

    I remember math professors constantly using questions like this in the vain attempt to edify students, failing to realize it was their own misguided understanding of human communication that was at fault.

  17. #17 Kevin
    December 29, 2006

    David-

    The MF and FM results are different. Lets say a man and a woman are standing in front of you, in sacks so you can’t see which is which. You must label them male or female. You know there is one of each. Since you don’t know which is which, you might label the female a male and vice versa, and you’d be wrong.

    That is why we must account for MF and FM. Sometimes, 1/3 of the time, Dog A and Dog B will be both be male. There’s only one way to do that. Sometimes Dog A will be male and Dog B will be female – 1/3 of the time. The last 1/3 Dog A will be female and Dog B will be male. There’s two ways to get the generic MF combination.

  18. #18 Dave S.
    December 29, 2006

    David Heddle says:

    Bill Buckner and Mookie Wilson had a free-throw competition. Since they live on different coasts, they just reported by phone how well they did.

    In the first round, Mookie made 40% of his shots, and Bill only made 35%
    In the second round, Mookie made 25% of his shots, and Bill only made 20%

    After the second round, Mookie declared himself the winner.

    Question: was he justified in doing so? If not, why? (You may assume total honesty in their reporting.)

    Mookie may have won, but we cannot know for sure using only the percentages given.

    Consider the following scenario –

    Round 1: Mookie = 40/100 (40%) Bill = 35/100 (35%)
    Round 2: Mookie = 1000/4000 (25%) Bill = 2/10 (20%)
    Total: Mookie = 1040/4100 (25.4%) Bill = 37/110 (33.6%)

    In this instance, Bill has the win. Unless they had agreed that only the percentage after each round was important.

    One of my favs:

    You have 12 identical small opague boxes, each containing a penny. A trickster comes along and takes a penny from one of the boxes and replaces it with some other coin that is either lighter or heavier than a penny. Using a balance scale and only 3 weighings, can you find the box containing the mystery coin?

  19. #19 Miguel Garcia-Blanco
    December 29, 2006

    Cute problem!

  20. #20 Chad Orzel
    December 29, 2006

    A shopkeeper says she has two new baby beagles to show you, but she doesn’t know whether they’re both male, both female, or one of each. You tell her that you want only a male, and she telephones the fellow who’s giving them a bath. ?Is at least one a male?? she asks him. She receives a reply. ?Yes!? she informs you with a smile. What is the probability that the other one is a male?

    Of course, the real question here is “Why are you such a sexist pig that you only want a male dog? What’s wrong with female dogs, hmmm?”

    (Forget the Bayesian approach, go with the Calvin and Hobbes approach…)

  21. #21 Jeff
    December 29, 2006

    I can see how the 1/3 is arrived at mathematically, and I see its justification. There’s no need to be dismissive because I’m defending the 1/2.

    In considering how this problem would play out in the real world, I’m convinced it’s a logical assumption that the bather would either examine one or both dogs and stop after finding a male. This makes 1/2 a valid answer. Maybe I’m bring extra stuff to the problem, but I don’t think the assumption is unreasonable or illogical.

    The coins example has the coins flipping simultaneously. Say one’s a penny, one’s a dime. Then it’s easy to see how we can end up with HH/HT/TH (if we disregard TT). But what if we flip the penny first and end up with H? Then the cases are HH or HT. This is how I see the dog problem. We know one dog is male, the bather looked. That can’t be changed. The only piece of additional information we can possibly get is the gender of the OTHER dog.

    I do realize that 1/3 is the proper mathematical answer but it doesn’t resolve with reality for me. It only works as a mathematical problem when given that “a” dog is male.

    Anyhow, that’s about all I have left to say :) I see how both answers work in certain circumstances. Thanks for giving this for us to toy with, Jason!

  22. #22 Billy
    December 29, 2006

    David,

    You asked how a “one-son” policy would affect the ratio of boys to girls in China. I’m curious whether my thinking is correct. Assuming that the probabilty of having a boy is 1/2, then 1/2 of all first children will be boys; 1/2 of all second children will be boys, 1/2 of all thirds, etc. Thus under such a plan the ratio of boys to girls would tend toward 1:1.

    (On the second question, Mookie is justified in declaring himself a winner if and only if he meets the requirements that he and Bill agreed upon for victory. If he and Bill failed to do so, then it wasn’t much of a competition.)

  23. #23 David Heddle
    December 29, 2006

    Billy,

    On the first problem, you are correct. Each birth is 50-50 and it doesn’t know or care about your policy for deciding when to stop.

    For the basketball question, Dave S nailed it. This is called “Simpson’s paradox.” To avoid it, equal sample sizes should be used for each round (you can see by his numbers that unequal sample sizes produced the paradox). This is important for drug companies: otherwise while each survey might show a drug is more effective for men than for women, the truth might be just the opposite.

    As for intuition, I agree with some of the posters here that probability can be highly counter-intuitive. The interesting thing is that if you have the math skills you can convince yourself, with confidence, of the right answer. But even more–the math will give you a feeling (perhaps an illusion) that the answer “feels right” in spite of your initial inclination to the contrary.

    Math rocks! Why, it’s a close second to physics!

  24. #24 JK
    December 29, 2006

    Perhaps we can illuminate the answer by considering a generalisation.

    Suppose we want 9 male puppies from a batch of 10 (apologies to Chad). The shopkeeper phones through and reports back that there are indeed 9 male puppies. What is the probability that the tenth is also male?

    Assuming that males and females are independently equally likely, you can see the distribution of puppies with probability on the vertical axis and number of males on the horizontal axis here:

    http://www.cse.msu.edu/~nandakum/nrg/Tms/Probability/chi.ht66.jpg

    (Hopefully this also accords with everyone’s intuition … at least after a little thought!)

    Now, we have some new information, namely that there are at least 9 males. How can this occur? There are two ways. Either the bather could have found 9 males and one female. There are ten ways in which this could happen. Alternatively, the bather could have found that all ten are male. There is one way in which this could happen. The probability of a tenth male is then 1/11. (This corresponds to the analysis that produces an answer of 1/3.)

    What about the analysis that produces 1/2 (which was my immediate “intuitive” answer)? This adopts a uniform prior probability distribution on the number of males. The immediate psychological plausibility of this is that the number of males is the variable that we care about, and to begin with we have no information.

    We could, in fact, carry through the analysis on the basis of this assumption. I think, however, that most people would agree that choosing sexes independently is a more plausible “ignorant” prior. The fact that this is not completely explicit does make this problem just a little less tightly stated than the original Monty Hall problem.

    PS If the pupies were indistinguishable quantum particles then in fact the second treatment would be apporpriate and in fact this has physical consequences which are used as an argument that all quantum particles of a given sort (e.g. all electrons) are strictly identical:

    http://en.wikipedia.org/wiki/Quantum_statistics

    PPS Now that this has been sorted out, perhaps the real professionals can turn their attention to this:

    http://en.wikipedia.org/wiki/Two_envelopes_problem

  25. #25 Stuart
    December 29, 2006

    “In considering how this problem would play out in the real world, I’m convinced it’s a logical assumption that the bather would either examine one or both dogs and stop after finding a male. This makes 1/2 a valid answer.”

    Ummm according to your logic that would be a 1/4 chance, not 1/2.

    It is 1/2 that bather looked at female then male, so 1/2 that the answer is ‘no’

    And one half that bather has not looked yet. Therefore it is a further 1/2 that the next dog is actually a male.

    Mathematically it is 1/3, according to your ‘real world logic’ it is 1/4. But never 1/2! :)

  26. #26 Mustafa Mond, FCD
    December 29, 2006

    Co-ed beagle bathing would be sinful. Therefore both beagles must be male.

  27. #27 MarkP
    December 29, 2006

    Jeff said: In considering how this problem would play out in the real world, I’m convinced it’s a logical assumption that the bather would either examine one or both dogs and stop after finding a male. This makes 1/2 a valid answer. Maybe I’m bring extra stuff to the problem, but I don’t think the assumption is unreasonable or illogical.

    Yes you are and yes it is and even so your logic is incorrect.

    First off, it is vitally important when calculating probablities that no new assumptions be made. They will often effect the calculations. There is nothing in the presentation of the problem that says anything about how the bather gets the information. Maybe he is going to examine them sequentially as you say. Or maybe he did that before the phone call and already knows the answer. Or maybe they are positioned in such a way that he can glance at both and get the answer. You don’t know, so you cannot make any assumptions about it. All you know is that he knows the answer, period.

    Second, even if your assumption is granted that he examines each dog and stops if he finds a male, so what? There are still 4 equally likely scenarios (FF, MF, FM, and MM), in three of those cases he is going to answer “yes” (twice after examining one dog and once after examining both), and in one of those three both are male, thus the answer remains 1/3.

    If you still doubt it, just take two coins, pretend they are dogs, and work through a few scenarios. It’s a small enough problem that I doubt you will have to run more than 12 trials before the answer becomes clear.

    Personally, I think this problem is more interesting for its psychological aspects than its mathematical ones. It’s a real test to see whether people are willing to override their instincts when the data suggests otherwise. Those who said people have bad instncts re statistics are right on. Even those of us who have training in statistics have shitty instincts. We just develop a new one: ignore your gut and calculate.

    But some people just cannot bring themselves to let logic and facts override their instincts when the instincts are strong enough. They are easy to spot, for they tend to make comments like this:

    Jeff said: I do realize that 1/3 is the proper mathematical answer but it doesn’t resolve with reality for me.

    Sorry Jeff, but that translates to “to hell with the facts, my gut knows better”. Trust the statistician: it doesn’t.

  28. #28 Jason Rosenhouse
    December 29, 2006

    Mike-

    You’re still making the error of thinking about individual dogs, instead of pairs of dogs. It is not that we are trying to use information about the sex of one dog to get information about the sex of the other dog. Rather, we are using information about the sex of one of the dogs to draw conclusions about the likelihood of different sex distributions for both dogs.

    Richard Wein-

    In my defense, I got the wording of the problem from vos Savant’s book. I merely copied it. I’ll try to phrase it more carefully in the future.

    Caledonian-

    I think you’re right about that. There have been a lot of studies showing that probabilistic reasoning is not something that comes easily to people.

    David D. G.-

    You actually mean “symmetric,” not “commutative.” “Commutative” is a word that gets applied to binary operations, like addition and multiplication. It means that the order of the two things being operated on is irrelevant. For example, addition is commutative because if a,b are integers, then a+b=b+a. Likewise, multiplication is commutative because a*b=b*a. But subtraction is not commutative because a-b does not equal b-a.

    As for the rest of your argument, think of it this way: Given a pair of dogs, the most likely scenario is that one is male and the other is female. Specifically, one male and one female is twice as likely as two males. Learning that one dog is male does not change that fact. That is, after learning that one dog is male, one male and one female is still twice as likely as two males. Hence the probabilities are 2/3 and 1/3.

    Jeff-

    Your comments raise an interesting issue. When mathematicians propose a problem like this, they generally don’t have a real-world situation in mind. In real life, if there were an actual shopkeeper discussing actual dogs with an actual bather, it would probably be fine to make added assumptions about how people are likely to behave and bring that to your analysis. But in a math problem the dogs, bather and shopkeeper are just devices for making certain abstract points.

    As an example, when mathematicians talk about flipping a coin, it is simply understood that heads and tails are equally likely. For real coins, that’s actually not true. The two sides of a quarter, for example, are not identical, and this results in a slight bias for one side over the other, even when the coin is tossed fairly. But in thinking about math problems that involve coin tossing, the coin is really just a device for discussing a process that has two, equally likely outcomes.

    Actually, your comment reminds me of a time I presented the Monty Hall problem to a group of students. As you know, the correct answer to the question is that you should switch doors. The common, incorrect answer is that it makes no difference whether you switch. So you can imagine my surprise when a student said, with great confidence, that you should stick with your first choice. I asked him why he gave that answer. He replied that you obviously had a gut instinct about your first choice, and you should always trust your gut.

    David Heddle-

    You realize that Bill Buckner and Mookie Wilson are baseball players, right?

    Jeff Chamberlain-

    Probability theory is often counterintuitive, so it can be difficult to get a handle on for most people. But I think it’s more a matter of hard work and practice, rather than innate differences in the brain.

    On the other hand, I would agree that mathematicians are sometimes insufficiently aware of the fact that they talk about things in a way that is different from how non-mathematicians talk about things. This shows in math textbooks, which tend to present mathematics in a stilted, Theorem-Proof-Definition kind of format, which in addition to being extremely boring often serves to make relatively simple ideas seem pretty complicated.

    One small example: In calculus classes it is common to ask a question like, “What properties does the function f(x) possess?” What is intended are things like: Is the function continuous? Is it differentiable? Are there any points where it is undefined? Does it have any asymptotes? Where does it cross ths x-axis? That sort of thing.

    For mathematicians these are very natural questions to ask. But for students the very idea that functions are the sort of things that have properties is very weird. To them a function is just some mathematical expression, like f(x)=x+2. End of discussion.

    hrln-

    Forgive me, but I’m not sure what you’re point is. Your problem about the two men walking down the street seems perfectly reasonable to me. And I’m not sure what you mean by saying that I did not define an event set.

    Miguel Garcia-Blanco-

    Glad you liked it!

  29. #29 David Heddle
    December 29, 2006

    Jason,

    You realize that Bill Buckner and Mookie Wilson are baseball players, right?

    C’mon, I live 40 minutes from Boston! They’re more famous that John Kerry. Besides, did you see the free-throw percentages I attributed to them? I couldn’t use basketball players (or, from the other end, chess players.)

  30. #30 Pseudonym
    December 29, 2006

    One of the interesting things about this is that the p=1/2 answer is true if the dogs obey Bose-Einstein statistics. Replace the dogs with neutrons and sex with spin up/spin down and you’re set.

    Just thought I’d throw that out there.

  31. #31 jeffperado
    December 29, 2006

    First, I must apologize for not reading all the comments, so if I’m duplicating something here, just ignore.

    Anyway, this is the flaw in the logic and why bayes theorem does not apply. We do not know if dog ‘x’ or dog ‘y’ was the one examined. Not only that, we cannot distinguish between the two as they are both independent. Since we have no dependent information and cannot distinguish between the two, then functionally x=y is the same as y=x.

    With that being the case, then bayes does not apply, and each event (sex of the dog) is independent and not biased by knowledge of other dogs.

  32. #32 MarkP
    December 29, 2006

    Jeffperado, the problem does not say “one dog is male”. If it did your conclusion would be correct. It also does not say only one dog was examined. Again, with problems like this it is crucial to base one’s answer only on the information given:

    “Is at least one a male??”
    “Yes!?”

    That is all we know. We know nothing about which dog is or is not male, nothing about how many dogs were examined. All we know is the number of males is either 1 or 2. And since we have 3 equally likely scenarios which match the criteria, FM, MF, and MM, the answer is 1/3.

    If you or anyone else still doubts this is the answer, then please take two coins out of your pocket and flip them. Is at least one heads? If no, record nothing. If yes, set the one that is aside, and answer: is the other one heads? Record the answer.

    Repeat at least 12 times, or until you are convinced, whichever is greater. It will take less time than your average post.

  33. #33 MarkP
    December 29, 2006

    Correction above:

    “one dog is male”

    should read:

    “one particular dog is male”

  34. #34 David D.G.
    December 30, 2006

    “You actually mean ?symmetric,? not ?commutative.? ?Commutative? is a word that gets applied to binary operations, like addition and multiplication. It means that the order of the two things being operated on is irrelevant. For example, addition is commutative because if a,b are integers, then a+b=b+a. Likewise, multiplication is commutative because a*b=b*a. But subtraction is not commutative because a-b does not equal b-a.”

    Ah, okay. Thanks for the math vocabulary lesson! Obviously I haven’t needed to use it for a long time.

    “As for the rest of your argument, think of it this way: Given a pair of dogs, the most likely scenario is that one is male and the other is female. Specifically, one male and one female is twice as likely as two males. Learning that one dog is male does not change that fact. That is, after learning that one dog is male, one male and one female is still twice as likely as two males. Hence the probabilities are 2/3 and 1/3.”

    Well, one of us just isn’t getting it here (probably me, I admit), but I don’t see how a male/female mix is twice as likely at all. The order of them doesn’t matter, really; you’re just talking about groupings, and order is immaterial. Grouping M/F and F/M as if it were two separate conditions is completely ARTIFICIAL. How would switching the relative positions of the puppies make any difference to the probability of their sex? It wouldn’t! And that’s the equivalent of what I see this doing.

    The poster above who gave the variation above (“Two people are walking down the street, one being a man; what’s the possibility that the other is also a man?”) put the matter exactly right, in my opinion. We have already ascertained that at least one of the dogs is male; we just need to know the sex of the other dog, and there are just two possibilities — male and female — so the probability of it being male is 50%.

    I’m sorry, Jason, I really don’t mean to be disrespectful, and I do acknowledge that this isn’t my field; but the way this whole problem is presented, I honestly can’t see how it makes sense to double the mixed-sex possibility and skew the probabilities here. When people first started bringing that up, I was sure you were going to point it out as wrong — so I was shocked when you actually followed exactly that line of thinking.

    The HH, HT, TH, TT coin-flip idea, by the way, seems equally absurd to be distributed like that; TH and HT are the SAME THING, a mixed-side combination. How can it be otherwise? Seriously?

    ~David D.G.

  35. #35 Dave S.
    December 30, 2006

    Well, one of us just isn’t getting it here (probably me, I admit), but I don’t see how a male/female mix is twice as likely at all. The order of them doesn’t matter, really; you’re just talking about groupings, and order is immaterial. Grouping M/F and F/M as if it were two separate conditions is completely ARTIFICIAL. How would switching the relative positions of the puppies make any difference to the probability of their sex? It wouldn’t! And that’s the equivalent of what I see this doing.

    Say a dog has 2 puppies one after the other. There is only 1 way the puppies can both be male. The first puppy has to be male (P=0.5), and the second puppy also has to be male (P=0.5). The overall probability is the sum of the two probabilities (0.5 x 0.5 = 0.25). There is also only 1 way the puppies can both be female. The first puppy has to be female (P=0.5), and the second puppy again has to be female (P=0.5). And again the overall probability is the sum of the two probabilities, (0.5 x 0.5 = 0.25).

    We have accounted for so far a probability total of 0.25 + 0.25 = 0.5. The only remaining situation (1 male and 1 female) must account for the remaining 0.5 probabilty (since the total must be 1.0). This makes sense mathematically, and it makes sense conceptually because there are 2 ways of having a male and a female puppy and not 1 as was the case for MM and FF. The dog can have the female first and then the male, or have the male first and then the female.

    Consider a slightly different question. A dog has 2 puppies. What is the probability they will be the same sex? What about the probability they will have a different sex? Clearly these cases are exhaustive, and the sum of the probabilities must be 1. I think part of the problem is psychological. It’s easy to visualize MM and FF as two separate categories (even though they can both be boxed into ‘same sex’), but much more difficult to think of MF and FM as 2 separate categories.

    Another way is to consider a pair of dice, one red and one green. You can only roll a 12 one way – both the red die and the green must show a 6. But you can roll an 11 in 2 ways – the red die could show a 6 and the green a 5, or the red die could show a 5 and the green a 6.

  36. #36 MarkP
    December 30, 2006

    David DG wrote: I don’t see how a male/female mix is twice as likely at all. The order of them doesn’t matter, really; you’re just talking about groupings, and order is immaterial. Grouping M/F and F/M as if it were two separate conditions is completely ARTIFICIAL.

    It is not artificial at all. The fact is you have two dogs. Distinguish them any way you like: larger and smaller, older and younger, spotted and not spotted, or even “first viewed” and “second viewed”. Any way you slice it, you still have two combinations, FM and MF. Now it is true that you don’t have to care about order: probability problems are posed all the time that dont’t. But in that case, instead of having two equally likely scenarios, FM and MF, you just have one doubly likely scenario of one male and one female.

    If you still don’t buy it, thnk of it this way: it is clear that the odds of getting MM = 50% X 50% = 25% correct? Ditto with FF, correct? That leaves only one other option, that being one male and one female, you know the total probability must be 100%, therefore the odds of one make and one female must be 50%.

    This is covered in any introductory text on probability, and isn’t the slightest bit controversial, so if you remain interested, I suggest a few minutes in your local bookstore.

    David D. G. wrote: The HH, HT, TH, TT coin-flip idea, by the way, seems equally absurd to be distributed like that; TH and HT are the SAME THING, a mixed-side combination. How can it be otherwise? Seriously?

    I repeat what I wrote above: this is too simple a problem to waste all this time arguing about. DO IT! Take two coins, flip away, and record your results. It won’t take as long as you took to submit your last post to see that the mixed result will appear twice as often as the other two. Accept the fact first, then we can work on understanding why it is so.

    If you want to convince yourself even faster, flip 10 coins. As I understand your view, you think you should get 5 heads and 5 tails just as often as you get 10 heads, correct? You won’t have to flip and record very long to see how wrong that is: you will actually get the 5 and 5 result about 252 times as often.

  37. #37 AJS
    December 30, 2006

    David D.G.,

    “HT” means the first coin came up heads, the second came up tails. “TH” means the first coin came up tails, the second came up heads. They aren’t necessarily the same thing — especially if the coins were a 2p and a 1, and you got to keep whichever one came up heads.

    Between them, these two outcomes account for 50% of two-coin tosses (the other 50% are split evenly between HH and TT). The most probable outcome is one of each, because there are two equally probable ways that you can arrive at that outcome.

    Going back to the dogs, I have drawn a tree diagram which shows the probability of each outcome:

    (click here to view)

    If that doesn’t convince you, I’m not sure anything will.

    Now, if we know that one of the events isn’t going to happen — in this case, two females — we have to multiply (because we want the probability of two events happening together: not-the excluded event and any one of the others) the probabilities of the remaining events by 1 / (1 – P) where P = the probability of the excluded event. The subtraction from one (1 = certainty) is because we want the probability of anything but that event; and we take the reciprocal of that, because we already allowed (in the initial calculation) for the probability that the excluded event might happen. In this case P = 0.25, so we multiply everything else by 1 / 0.75 = 4/3, as the new diagram shows:

    (click here to view)

    That’s why it’s 33%, and not 50% or 25%.

    Incidentally, here’s a trick for estimating probabilities of two combined events (especially events with uneven probabilities) graphically: Draw a square, say 10cm. by 10cm., and divide it horizontally according to the outcomes of the first event, the width of each portion being proportional to the probability of that outcome. For a 6-sided die, that means six columns each 1.6…cm. wide. Then divide it vertically according to the outcomes of the second event. Each subdivision corresponds to a combined outcome, and its area divided by the total area represents its probability: the most probable outcomes will be represented as the largest are If you are excluding any particular combinations of events, subtract their areas from the total before you perform the division.

  38. #38 Jeff Chamberlain
    December 30, 2006

    JR — Your implication (very diplomatically put) is that failure to understand probability can be cured with more hard work and practice. Maybe so.

    BUT: I, myself (and, I’m pretty sure, others), have put in a good deal of hard work and practice trying to understand probabilities, with limited success. I’m also pretty sure many people have put in less effort and don’t understand probabilities. If you are correct, then it seems to follow that to understand proabilities will require more work than many people will be willing or able to put in.

    Assuming that people are not willing or able to put in sufficient work to understand probabilities, at least some of them will nonetheless understand that probabilities are counterintuitive (or beyond us, for whatever reason). Many of those people will be sophisticated enough not to rely on their intuitions when confronted with a probability question. They may, instead, try to rely on experts. If I identify a probability question, I may well say something like “OK, I’m not competent to resolve this so I’ll find someone who is and proceed based on their answers.”

    How do we determine who such experts are and whether their answers are authoritative? Every time I’ve seen “Monty Hall” type problems discussed, there seems to be substantial dispute, including among people who seem to have “expert” credentials.

    Here’s another way to ask this question. In the beagle problem, for example, you tell us that the answer is 1/3. Is this a “fact,” or is it your “opinion?” Would “all” mathematicians (who are not crackpots) agree with you? Can we amateurs (smart enough to look for accuracy in resolving probability questions) rely on “most” working mathematicians coming up with the same answer (and is there a way (for us amateurs) to identify those working mathematicians who can be relied on)?

  39. #39 Kevin
    December 30, 2006

    David DG-

    OK if you don’t understand why MF and FM are different, then yes, you will never understand why 1/3 is the right answer. The only way I guess to prove this to you is start flipping coins. 1/4 of the time, you’ll get HH. 1/4 of the time you’ll get TT. And 1/2 of the time, you’ll get HT. Do it. Go. When you see it play out, read the explanations of why MF and FM are important. Sorry but it seems that empirical evidence is the only way for you to see this. Flip, flip, flip.

  40. #40 Kevin
    December 30, 2006

    Jeff C-
    the 1/3 answer, based on the point (and not the nitpicky deconstruction) of the problem, is a factual answer. Many people have intuitive failures when it comes to probabilities – go to Vegas for real life examples of this. And honestly, an amateur can do the beagles problem on their own. Flip coins, ignore any TT outcomes, and count.

    As for harder statistical problems, well you just have to trust those who you consider trustworthy. I don’t know crap about how to build a plane, but I fly from time to time reasonably sure they got the “right” answer on how to put an engine together. That’s just the way life is.

  41. #41 MarkP
    December 30, 2006

    Jeff C: How do we determine who such experts are and whether their answers are authoritative? Every time I’ve seen “Monty Hall” type problems discussed, there seems to be substantial dispute, including among people who seem to have “expert” credentials.

    It is difficult, granted. Look for who can show more than one way to get their answer. Beware those whose answers seem to change the problem a lot. The rest is mostly standard practice for discerning a real expert from a fake one in any area of life: look for consistency with experts in other fields when those fields overlap, an ability to express their views abstractly and generally, and I’m sorry, yes, real life credentials. Yes, people with credentials can still be wrong, and yes they sometimes argue amongst themselves, but when its a credentialled person in the field arguing with someone who isn’t, about that field, the credentialled person is gonna be right 99% of the time.

    Here’s another way to ask this question. In the beagle problem, for example, you tell us that the answer is 1/3. Is this a “fact,” or is it your “opinion?”

    It’s a fact. The reason is because you can lay out all the possible scenarios and actually count through them all. We have the formulations too, but we really don’t need them.

    Would “all” mathematicians (who are not crackpots) agree with you? Can we amateurs (smart enough to look for accuracy in resolving probability questions) rely on “most” working mathematicians coming up with the same answer (and is there a way (for us amateurs) to identify those working mathematicians who can be relied on)?

    For something this simple, yes, especially after discussion. For the record I have a BS in math with an emphasis on statistics and probability, and a professional designation as an actuary. Yet my instincts were all wrong with this problem, until I had the other view explained, and then I understood my error. That’s the thing. We can all make mistakes, but usually when an expert errs in his own field, and its pointed out to him, he’ll recognize it right away.

    But as I stated earlier, these sorts of problems have remarkable psychological content. I once saw a family meal turned into a raging shouting match of an argument that didn’t end for weeks over the Monte Hall 3-Door problem. I think it is because it is so counterintuitive, every fiber in your being wants the answer to be one thing, and sometimes even looking at the scenarios and ironclad logic of why the answer is something else just won’t persuade.

  42. #42 Caledonian
    December 30, 2006

    My intuition is that we have a whole series of pragmatic mental modules that are designed to evaluate real-life situations involving probability and apply some basic rules-of-thumb to produce useful and beneficial results. When we try to apply correct mathematical thinking, our reason conflicts with those instincts, and the result is a mental wrestling match that’s less useful than either the heuristics or reason alone.

  43. #43 Jeff Chamberlain
    December 31, 2006

    My real question, I suspect, is that I don’t understand why I don’t understand probability. JR may be right that it’s just not enough hard work, although I’ve put more effort into trying than I’ve had to exert to understand things which are presumably as difficult.

  44. #44 David D.G.
    December 31, 2006

    My thanks to Dave S. and Mark P. (and, of course, to Jason) for so patiently trying to explain this situation to me. The (to you) basic idea of mulitiplying individual probabilities to get overall probabilities totalling to unity did not get through to me until this point (if it even was brought up earlier — I’m not sure, as I kind of had my eyes glaze over during mathematical specifics in earlier posts). I was substituting what seemed like “logical” reasoning for the actual math.

    So, okay, I get it now — barely. I see now how the MF and FM are not quite the same thing for purposes of probability. It still feels incredibly weird to try to fit that thought into my head — my mind does NOT want to accept such a counterintuitive notion! — but I do follow the reasoning, finally, at least after a fashion.

    Again, thanks to everyone for being so patient with me.

    ~David D.G.

  45. #45 Rev. Dan
    December 31, 2006

    Thanks for this post. The comments/discussion have been really interesting to read, and I needed a mental kick in the pants this morning.

    @Jason:
    > You realize that Bill Buckner and Mookie Wilson are
    > baseball players, right?

    Baseball players can’t shoot free throws? :)

    > First off, it is vitally important when calculating
    > probablities that no new assumptions be made.

    I mean no ill-will or snark, but I think this is an example of adding an assumption to the question. I think we’re all prone to doing that, regardless of our degree of learnedness. (It’s also/primarily an “aside comment” that’s quite funny.)

    > I’m sorry, yes, real life credentials. Yes, people with
    > credentials can still be wrong, and yes they sometimes
    > argue amongst themselves, but when its a credentialled
    > person in the field arguing with someone who isn’t, about
    > that field, the credentialled person is gonna be right
    > 99% of the time.

    I agree with your statments up until the bit about “credentials.” Personally, I think degrees and titles are generally pretty meaningless. Empirically, I know lots of “uneducated folks” (meaning that they have not received a degree from an educational institution) who can out-think and out-reason people with degrees. “Professor” or “Reverend” or “insert_title_here” mean little to me personally in terms of valuation of what the person has to say. I think a person’s body of work should be generally evaluated on its own merits. Anti-reason and Anti-logic seem very self-apparent, and clearly indicate cruft that should be disregarded.

    That said though, you can bet I’m going to go to the Specialist who graduated with Honors from a Medical School with a good reputation before I go to a witch doctor. I basically disagree with the the “99% of the time” part, as I feel that’s grossly overstated. Common sense suggests that someone who has applied intellectual and academic vigor and honesty to their pursuit of knowledge in their field will know more and have more valuable insight than someone who has done no study/research. My point is that a degree doesn’t inherently indicate that the recipient of the degree has actually earned it. It’s sad that academic institutions will actually award degrees to people who are good at showing up and taking tests (or whose parents donated the funds for the new library) but can’t differentiate between their head and their anus.

    (I’m not asserting this is the case with anyone here… I’m disagreeing with the inference/implication that the opinions of people with degrees should be valued more highly than those without. It’s the 99% part that chaps my hide a little.)

    My friend and I had a discussion about the dog problem this morning… neither of us intuited the correct answer. I think the big “money question” that I’ve seen posed in the comments is “why do we human beings so strongly resist that which can be logically proven and get so worked up when we learn that we’re provably wrong?”

    One of the things I most admire about scientists like Richard Dawkins is that he basically says (total paraphrase/rendering) “there’s a chance I’m wrong, because there are plenty of things that we just don’t and can’t *know.* The overwhelming evidence, and Reason, suggest that…” I like that degree of intellectual honesty.

    I don’t mean to subvert the actual discussion into philosophical navel-gazing, but these types of logic questions/”puzzles” typically turn into “why are we so unwilling to give up comfortable false thinking?” to me. It seems fitting, on the last day of the year, to be challenged to think about this, and thanks Jason and commentors for that. :)

  46. #46 Caledonian
    December 31, 2006

    I think the big “money question” that I’ve seen posed in the comments is “why do we human beings so strongly resist that which can be logically proven and get so worked up when we learn that we’re provably wrong?”

    We very likely have built-in ways of handling such problems without reason and are disturbed when our reason conflicts with the built-in methods.

  47. #47 Rev. Dan
    December 31, 2006

    @Caledonian: I think we both agree with that thesis, and there are numerous examples/plenty of anecdotal evidence to support that there’s a “built-in way” or mechanism that causes this discomfort when confronted with reality.

    I guess what I’m trying to get at is “what specifically is that mechanism?” and “why?” Is it pride or arrogance? Is it just fear of uncertainty (ie.- “if I accept that I’m wrong about this, then what else am I wrong about”)?

    I also note that the subject is tangental and off-topic for a blog whose focus is on evolutionary science vs. “Intelligent Design” (or whatever the Creationist “flavor of the day” is), and it’s truly not my intent to be a rude guest. However, I have a personal investment in understanding this (and overcoming it), and would greatly appreciate any pointers.

    Perhaps a reader here might be able to direct me to a good Cognitive Science blog or the like (or suggest [an]other resource[s])?

  48. #48 Hamboy
    December 31, 2006

    I actually went with a 2/3.
    Since you are basically just saying is X (works for Y too) a male and from your list of three possibilities male appears twice.

    However my first thought was.
    Assuming the person bathing them wasn’t just told by someone else that one is a male. Then they are a sod, for not pointing out a male. lol

  49. #49 David M
    January 1, 2007

    In the macroscopic world, 1/3 is correct. In quantum mechanics, 1/2 is correct. As silly as the 1/2ers sound, I find it fascinating that such reasoning may actually be valid for “indistinguishable” particles. Hey what if the two dogs are identical twins or clones? :)

  50. #50 Hamboy
    January 1, 2007

    Forget my 2/3 comment. I miss read it.
    It was news years day and I had a hang over. Was a good night.
    Don’t let the time stamp fool you. We are nearly a day a head of you.
    I do have a problem with the way the question was worded.
    It gives the impression that you are only looking at one dog and guessing whether or not it is a male. 1/2 chance. It’s the use of the phrase “the other dog”. As in we know what sex one dog is, what is the sex of the other one. Instead of saying, “One of them is male, can you guess which one”

  51. #51 MarkP
    January 1, 2007

    Rev. Dan said [of my claim that no new assumptions be made to the problem when calculating probabilities]:
    I mean no ill-will or snark, but I think this is an example of adding an assumption to the question. I think we’re all prone to doing that, regardless of our degree of learnedness.

    How is adding an assumption to not add assumptions?

    Rev. Dan said: I agree with your statments up until the bit about “credentials.” Personally, I think degrees and titles are generally pretty meaningless. Empirically, I know lots of “uneducated folks” (meaning that they have not received a degree from an educational institution) who can out-think and out-reason people with degrees.

    Well, first off, that’s not the point I was making. I was speaking specifically and only about a discussion within that specific field between someone credentialed in that field and someone not. History is full of experts in their field being total doofuses outside it.

    However, I suspect you have overestimated the victories of the folksy, because circular analyses are common when scoring those debates. When folksy guy Bob decides that folksy guy Jim out-thought and out-reasoned credentialed guy Sam, it’s often only because Bob has the same flawed views as Jim on the subject. For example, think of the many creationists who leave debates convinced the creationist showed that fancy-degreed scientist a thing or two, merely because the creationist debater appealed to the flawed views of his audience. By contrast, how often have you ever seen Mr. Folksy PROVE Mr. Degree wrong? My bet is not very.

    I have studied many areas of legitimate knowledge in depth, and never, not once, have I ever come away from those studies thinking “Gee, that subject was just as simple as I thought it was. My neighbors down the street know just as much about it as the experts”. No, 100% of the time, what I come away thinking is “damn, there is a lot more to that than I thought, and the folksy views of my neighbors are completely off base and often flat stupid.” My 99% comment might be overstating the matter a bit, but it’s a sight closer to the truth than your claim that “degrees and titles are generally pretty meaningless”. I mean really, what do you think they do in colleges, just enroll and pass people at random? That’s what “essentially meaningless” would require. Do you really think your average math major and the average person on the street know about the same amount of math? I’m pretty damned sure they don’t.

    Rev. Dan said: why do we human beings so strongly resist that which can be logically proven and get so worked up when we learn that we’re provably wrong?

    Sometimes its because we are emotionally invested in the idea. Sometimes its because we have based a lot of our life on that assumption. Some people think their gut just can’t be wrong. It’s all part of why it is so important to always be on guard for error, to subject your views to others for analysis, and to keep in mind what sort of outcome would warrant rethinking one’s views. That’s why science is so effective: it assumes from the get-go that we are biased and not always objective.

  52. #52 Curt Sampson (cjs@cynic.net)
    January 1, 2007

    I think that the question, “why do we human beings so strongly resist that which can be logically proven and get so worked up when we learn that we’re provably wrong?” is not at all off-topic for an evolution vs. ID blog. It appears to me to be the core of the whole problem. Those who favour ID over evolution do not do so due to lack of overhwelming evidence for evolution. ID proponents, for the most part, are not people who merely don’t have enough knowledge of philosophy of science and/or the natural world that, if they acquired this, they would change their minds.

    Right now, we don’t battle IDers in order to convince them; we do it merely to slow and stop the spread of the disease. That’s why it’s a rather depressing battle.

  53. #53 Rev. Dan
    January 1, 2007

    @Mark:

    >> Rev. Dan said [of my claim that no new assumptions be made
    >> to the problem when calculating probabilities]:
    >> I mean no ill-will or snark, but I think this is an example
    >> of adding an assumption to the question. I think we’re all
    >> prone to doing that, regardless of our degree of learnedness.

    > How is [it] adding an assumption to not add assumptions?

    I meant that the statement “You realize that Bill Buckner and Mookie Wilson are baseball players, right?” is ironically (and humorously) adding an assumption… the assumption being that the names are relevant information to the solving of the problem.

    I fully agree that we need to challenge our assumptions and presuppositions, regularly, and I’d assert in every area of our lives. If we can’t “kick the tires” (please forgive my “folksy” expressions… I’m “folksy” :) ) there’s not much assurance that the car we’re riding in is “safe.” If one can’t challenge one’s own thinking or beliefs, then they’re likely thinking or beliefs that we shouldn’t be holding as precious.

    > My 99% comment might be overstating the matter a bit, but
    > it’s a sight closer to the truth than your claim that
    > “degrees and titles are generally pretty meaningless”.

    My overarching point was that I think that “99%” was quite overstated. Upon reflection, it seems that my response was also overstated, though I did use the term “generally” (meaning broadly) a bit excessively, and quite intentionally.

    My experience with folks with advanced business degrees (or technology degrees/certifications) provides me with empirical evidence that just because someone has a piece of paper which indicates that they were able to sit through x amount of classes, submit y amount of homework, and scored z well on standardized testing doesn’t necessarily mean that they’re able to think or reason, let alone that they actually studied and applied themselves to the domain of knowledge in which they’ve received a degree.

    I have personally witnessed the bad business decisions made by one-too-many morons with business degrees from Harvard. I don’t mean to assert that all Harvard graduates are morons (I know smart Harvard graduates as well), but rather that a degree from an educational institution shouldn’t be valued as exclusive evidence of degreed person A’s superiority to non-degreed person B… provided person B has studied the subject matter and applied themselves with intellectual and academic vigor. You don’t have to have a degree to be an expert in every specific domain of knowledge. The Arts are an area where this is particularly obvious. Did Picasso or Goya have a Fine Arts degree? If they didn’t, does that mean they were just folksy painters?

    One doesn’t need to have a degree to prove that one has learned anything.

    I think it’s sickening that universities will award degrees to the uneducated, and it’s beyond discouraging to those who *have* applied themselves to their field of knowledge and worked their asses off towards mastery of domain knowledge in their field. Degrees are a piece of paper which are devalued by the reality that colleges *will* award degrees to those whose parents have donated the new library or science wing, and/or those who are just simply good at taking exams.

    I can understand why you’d be at odds with my statement because it appears your focus is primarily on mathematics and science. Further, I see that I could definitely have provided more context, and that your focus is reasonable in the context of this being an entry on statistics on a science blog. It seems reasonable to assert that the actual percentage of folks with mathematics, engineering, etc. degrees would be far more likely to be able to capably demonstrate their skill and expertise than those who have not pursued their expertise in an academic environment.

    > However, I suspect you have overestimated
    > the victories of the folksy,

    I suspect that you’ve misinterpretted what I’ve written, didn’t read it completely (I’m verbose), or are making assumptions regarding my intent that are not in alignment with what I meant. I feel that the paragraph that follows the statement that you object to provided a bit more context. Does the bit I added below make this more clear/less objectionable?

    > “Professor” or “Reverend” or “insert_title_here” mean
    > little to me personally in terms of valuation of what
    > the person has to say [in and of itself, exclusively].
    > I think a person’s body of work should be generally
    > evaluated on its own merits. Anti-reason and Anti-logic
    > seem very self-apparent, and clearly indicate cruft
    > that should be disregarded.

    If Richard Dawkins wrote a book which denounced science as being illogical and argued for unsupported adherence, faith in, and allegiance to the tenets of faith in the one true god, the Flying Spaghetti Monster, he’d quite correctly be denounced as being a quack. There’d be no logic, reason, or body of scientific evidence to support those claims. The fact that Dawkins has degrees from an institution don’t make a difference in this (fabricated) case.

    Your statement seems to connote that there is no middle ground between those who have a degree and those who are backwoods “hicks” or are “folksy.” My point is, and remains, that just because someone has a piece of paper doesn’t necessarily mean that they actually are worth listening to. There are plenty of “scientists” who hold degrees and are completely full of crap (Dr. Michael Behe’s name springs immediately to mind).

    Personally, I could care less what degrees Richard Dawkins holds. What’s compelling about Dawkins’ work is that it’s incredibly clear and concise. He’s remarkably intelligent, well-spoken, and most importantly his statements withstand intellectually honest scrutiny and there’s an overwhelming body of evidence to support his statements and reasoning. I personally appreciate Dawkins because he’s quite capably championing logic and reason in an age of anti-intellectualism and anti-reason.

    >> Rev. Dan said: I agree with your statments up until
    >> the bit about “credentials.” Personally, I think
    >> degrees and titles are generally pretty meaningless.
    >> Empirically, I know lots of “uneducated folks”
    >> (meaning that they have not received a degree from
    >> an educational institution) who can out-think
    >> and out-reason people with degrees.

    > Well, first off, that’s not the point I was making. I was
    > speaking specifically and only about a discussion within
    > that specific field between someone credentialed in that
    > field and someone not. History is full of experts in their
    > field being total doofuses outside it.

    With that clarification, I’m again in almost complete agreement with you.

    > History is full of experts in their field being total
    > doofuses outside it.

    Wanna talk about religion? (No, not really… I’m just kiddin’) :)

    > However, I suspect you have overestimated the victories of the
    > folksy, because circular analyses are common when scoring those
    > debates. When folksy guy Bob decides that folksy guy Jim
    > out-thought and out-reasoned credentialed guy Sam, it’s often
    > only because Bob has the same flawed views as Jim on the subject.
    > For example, think of the many creationists who leave debates
    > convinced the creationist showed that fancy-degreed scientist
    > a thing or two, merely because the creationist debater
    > appealed to the flawed views of his audience.

    What you’ve presented is a scenario in which we both agree that Bob and Jim are demonstrating willful ignorance and profound stupidity. This is a good example of what you’re trying to communicate, but is not a good example of what I meant.

    An example of what I meant, from my personal experience: I went to school to become a Microsoft Certified Systems Engineer. Shortly afterwards, I got a job wherin I could use the skills that I’d learned. I had a great degree of technical knowledge about how to use the particular technologies, but found myself in a situation where regardless of what I’d been taught, regardless of the recommended best practices that we’d learned, regardless of the hands-on experience I’d gained, I was thoroughly unable to get a particular service to talk to another. I asked my team lead for help, and he recommended doing something that made absolutely no sense whatsoever. Figuring there was no reason not to try it (I’d tried everything else I could think of), I did… and it immediately worked. My team lead had no formal training whatsoever in the specific technology (and in fact, we were shocked at some of the remarkably stupid things he did to/on the network), but he was able to suggest a method that worked. I had certifications and theoretical knowledge (which I worked very hard for), whereas he did not. If I’d arrogantly assumed that he was flat-stupid because he was uncertified, I’d likely have not asked him for help, and would have been defeated by that problem (which would have made *me* a moron).

    Are you assuming that I’m a Creationist or an advocate of Intelligent Design because “Rev.” is in front of my name? Do you assume that anyone who holds a title or degree which has a religious connotation is a “flat stupid,” “folksy” moron? If so, why are you bothering to address a single word in my direction?

    > However, I suspect you have overestimated the victories of
    > the folksy…

    > I have studied many areas of legitimate knowledge in
    > depth, and never, not once, have I ever come away from
    > those studies thinking “Gee, that subject was just as
    > simple as I thought it was. My neighbors down the street
    > know just as much about it as the experts”. No, 100% of
    > the time, what I come away thinking is “damn, there is
    > a lot more to that than I thought, and the folksy views
    > of my neighbors are completely off base and often flat
    > stupid.”

    This smacks of arrogance, and the tone comes across to me as patronizing, perhaps because it’s overstated and oversimplified. If you’re suggesting that those who haven’t studied what you have are, by default, ignorant hicks who should bow to your profound knowledge, I think you’re being myopic and narrow-minded. If I’m rendering that incorrectly, then well, I’m obviously mistaken.

    Knowledge is like a balloon… just as one blows air into the balloon and the surface area increases, so adding facts and knowledge to one’s learning expands the awareness of the fact that there’s more to learn. The more we learn, the more we realize how much more there is to learn and how relatively little we actually know. Humility, not arrogance, would seem to be the natural byproduct of this process.

    > That’s what “essentially meaningless” would require.
    > Do you really think your average math major and the
    > average person on the street know about the same amount
    > of math? I’m pretty damned sure they don’t.

    It seems to me that the bottom line in this exchange is that my lack of willingness to automatically defer to someone who has a degree as being exceptionally more knowledgeable/correct without evaluating the work they produce or what they have to say offends you for some reason. Perhaps you’re mistakenly assuming that my statement means “there’s no point in honest academic pursuit of knowledge” or “there’s no value in having a degree.” I mean neither. However, I do not, by default, feel that someone with a label in front of their name automatically merits superior treatment or should gain instant credibility over someone who does not.

    One of the things that I observed in this comment thread was that someone was unwilling to accept someone else’s answer based on faith that the poster’s answer was correct… they demanded proof, and that proof was given and repeated until that person “got it.” That’s what prompted me to post in the first place. I admire the folks who understand things and will bear with those who do not until they understand as well.

    > That’s why science is so effective: it assumes from
    > the get-go that we are biased and not always objective.

    In my folksy venacular, you’re now preaching to the choir. :)

  54. #54 Toby
    January 2, 2007

    Lets name the two dogs – Binky and Berky.

    Case 1: Binky is Male, Berky is Male
    Case 2: Binky is Male, Berky is Female
    Case 3: Binky is Female, Berky is Male
    Case 4: Binky is Female, Berky is Female

    Each case has probability 1/4 as in JR’s development above.

    Answer: The information “At least one dog is Male” excludes Case 4 only, so the answer 1/3 is correct, whether you work it out by assigning equal probabilities, or by Bayes Rule.

    Alternative Information: If the information given was “Binky is Male”, not “At least one is Male”, that excludes Cases 3 and 4 so the probability that the other is male would then be 1/2.

    The Alternative is like the Monty Hall problem – the probability of the second dog being male goes from 1/4 to 1/2 because of the extra information given by naming the dog (like specifying a door).

  55. #55 Pseudonym
    January 2, 2007

    David M: Actually, 1/2 is only correct for bosons.

    You’re right that if the dogs obey Bose-Einstein statistics, MM, MF and FF are the only possible states, and they’re equally likely. MF and FM are not distinct states since the dogs are indistinguishable.

    However, if the dogs were fermions, then the answer would be 0. There is only one valid state: MF. (FM is observably the same as MF, but it has an opposite-sign wavefunction.) In Fermi-Dirac statistics, MM or FF states would disobey the Pauli exclusion principle.

    Would anyone care to suggest an experiment to determine the dogs’ spin?

  56. #56 Aron
    January 2, 2007

    The “Nebulous Neighbors” problem in the Rosenthal paper you linked to reads very similarly to the “baby beagles” problem, but the answer is 1/2. The difference seems to be that in the “Nebulous Neighbors” problem, the reader can identify one child as “the child by the window”. Just as if (as Jason noted) the reader was told that the beagle by the door was male. Ultimately, parsing the language to identify the key assumptions is as difficult as the statistics.

  57. #57 MarkP
    January 2, 2007

    Rev. Dan, a fine effort, and appreciated. We are in 95% agreement, and I see no reason to pick nits. So I will comment on but a few things:

    One doesn’t need to have a degree to prove that one has learned anything.

    Agreed. My point was aimed at the mytholocal “all else being equal scenario”. Obviously knowledge can be gained in places other than formal arenas. If I may wax folksie, Bob Dylan was right when he said “you don’t need a weatherman to know which way the wind blows”. The degree merely earns a benefit of the doubt, subject to revision, and in direct proportion to how rigorous is the system that granted the degree. A PhD in mathematics from Dartmouth means a lot, a BS from Podunk State in Wicker Engineering Arts, not so much, and a lot in between. I just focused on the extremes to make the examples more vivid. It’s a mathematician’s vice.

    I think it’s sickening that universities will award degrees to the uneducated, and it’s beyond discouraging to those who *have* applied themselves to their field of knowledge and worked their asses off towards mastery of domain knowledge in their field.

    Then surely you understand why some of us who worked our asses off earning a challenging degree might get a wee bit touchy when someone suggests it is next to meaningless.

    Degrees are a piece of paper which are devalued by the reality that colleges *will* award degrees to those whose parents have donated the new library or science wing, and/or those who are just simply good at taking exams.

    I’d say your first point is invalidated due to the populations being different by orders of magnitude. I’m sure the number of people who get a degree by a clerical mistake is not zero. So what?

    As to your second point, I have immense skepticism that there is such a thing as people who can consistently pass exams on material they do not understand (at least those given by legitimate institutions), and even less confidence that there are great masses of people who do understand the material but for some reason forget it when confronted by pencil and paper. It sounds exactly like the kind of rationalizations psychics go through when they fail tests. “I can only do it when you don’t watch me”.

    I’d say its far more likely the flunker wasn’t up to snuff at whatever is being measured, and is just having a hard time accepting it. It’s a very bad habit we all must battle, and IMNSHO is an excuse that should only be bought when the evidence is overwhelming. It is too easy to rationlize away all one’s falures this way, and then you improve on nothing.

    This smacks of arrogance, and the tone comes across to me as patronizing, perhaps because it’s overstated and oversimplified.

    Not at all, its actually the opposite. I just wrote poorly and forgot to explicitly include my former self among the unwashed. I thought it was understood. The point was that I’ve never studied something in depth, found nothing there, and came away with all the same opinions I held when I started, and so therefore it is silly to act like in-depth study doesn’t add knowledge.

    The more we learn, the more we realize how much more there is to learn and how relatively little we actually know. Humility, not arrogance, would seem to be the natural byproduct of this process.

    I have said this almost verbatum many times. I think where we differ is that, generally speaking, I believe it is those who lack credentials that need to hear this lesson most, not those who have them.

  58. #58 Rev. Dan
    January 3, 2007

    @Mark: Much thanks for your reply, it’s greatly appreciated.

    I have the notion/sense that were we kicking back having a brew in pub, we’d have come to an appreciation that we’re basically in agreement with quite a bit less prose/effort.

    95% agreement is pretty damn good… how do we move from here and try to help folks who are at 5% get closer to at least 50%? :)

    For me, one the more valuable takeaways from this exchange is that we all think our writing is perhaps more clear and precise than it is (and therefore our position/perspective should be obvious to the reader though it may not be), and we could all stand to work a little harder there. I certainly need to work on that more, and thanks for helping me see that.

    > I just focused on the extremes to make the examples more
    > vivid. It’s a mathematician’s vice.

    It’s a vice of artsy-humanities-types too. :)

  59. #59 David M
    January 4, 2007

    Oh thanks Pseudonym, my university failed to teach the fundamental quantum mechanical destinction between bosons and fermions. Anyone faintly familiar with the topic knows this distinction, but these pompous details are irrelevent and discouraging to the layman, at whom I was targeting a conceptual fascination with the counter-intuitive(for most!?) nature of the quantum realm.

  60. #60 Stuball3D
    January 9, 2007

    Very interesting stuff. Would the probabilities be altered by when the question was asked? If the washer was asked before s/he had seen either dog then it would be 1/3 m:f after having seen the first dog. But if s/he were asked after s/he had seen the first dog then the probability would be 1/2, right? It seems like the probability changes by whether the washer sees them both at once, or one after the other.

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