Tuesday, April 3, 2012

This Should Be Easy

Find the error in Gene Callahan's reasoning. He posts:

"There's a 0% chance that X will happen." 
Most people interpret that to be equivalent to, "X will never happen." 
But that isn't so. An example: If one were to toss an infinitely fine-pointed dart at the real number line and hit a spot between 0 and 1, for any particular number in that range, there is a 0% chance you will hit it. But you *will* have hit *some* number, so even though there was a zero percent probability that number would be hit, it was hit.
Answers in comments, please.


  1. He is playing with the concept of absolute 0 and 0 as a limit. The reasoning will work well at amazing intellectual wannabes with low mathematical skills, but its rather stupid.

    1. It is absolutely standard probability theory.

    2. Sorry for the late reply, I did not realize this post had generated such controversy.

      Im not discusing the math concepts used so your response makes no sense (again). What Im saying is that you are playing with the concepts of absolute cero and cero as a limit to give the impression of writting something paradoxical and very meaningful when your comment is neither. To someone who understand the mathematics of what you are saying the comment is only as pretentious as meaningless.

  2. 0.00...001 does not equal zero.

  3. There is an infinity of numbers on any subset of the number line, so the odds of hitting one of them is 1/inf, or "0" as it's commonly known.

  4. this is just foolish. first of all saying that you are throwing an infinitely small dart is a ridiculous notion to start his reasoning off with. as well,throwing this dart, there was no o% possibility... there was one out of whatever.
    i get it, he's trying to be clever by playing with infinity. there are probably other reasons this is wrong, that's just how i saw it.

  5. Infact you will hit no number as the point of the dart is infinitely small and the separation between the numbers is infinitely small (as there are infinite numbers between 0 and 1). So the chance of hitting the number are zero and no number will be hit. Gene callahan must be a Keynesian, using confusing analogies to arrive at misleading conclusions.

  6. In order to hit a number, the dart would have to have a finite width. Otherwise, the dart would keep flying into the number line until a section of the dart with finite width hit said line. Or the whole dart is infinitely fine, and would pass right through.

  7. 0/n =0

    Is there some other math at work here?

  8. As the number of possible events or points increases within a set, the probably that anyone event or point will be selected approaches zero. If you have an infinitely small point dart, there is a zero chance it will hit any given point. Limit as z goes to infinity of 1/z equals 0. The problem here is that he bounded the set between 1 and 0.

  9. Well, the dart can hit no number because on a perfectly continuously real valued line, the number of positions after the decimal are infinite and you would spend eternity trying to determine the number you actually hit. You would have to "discretize" the values on the line to some degree and then, of course, you would be approximating and not actually stating the value hit.

  10. He must of been drunk during the probability portion of statistics and probably never took critical thinking.

  11. There is a 100% chance that Gene Callahan is lost in semantics.

    If some event occurs, by definition it had a non-zero probability before the event happened.

    This does not imply that the probability is not so infinitesimally small that it can be rounded to 0% whenever it is convenient for the argument of any given economist.

  12. He could very easily entirely miss the number line, thus not hitting any number at all.

  13. If the number can be hit, then the chances of hitting it are not 0%.

  14. Another thought is that no matter how infinitely fine the dart is, it still takes up mass, thus if you did indeed hit the number line, the dart would fall on a range of numbers, not a single number.

  15. Mathematically speaking I think he is correct if you think back to the theory of the limit. How can you hit 0 or 1 if you can infinitely divide the space between 0 and 1?

  16. excuse me, that should have been "still has mass" or "takes up space". For some reason I combined the two which resulted in "takes up mass". Doh!

  17. He assumes there is such a thing as a "zero percent chance"which is not true. There is always a possibility that X will happen, even if it is very small.

  18. That means that...
    our whole solar system...
    could be, like...
    one tiny atom in the fingernail
    of some other giant being.
    This is too much!
    That means...
    one tiny atom in my fingernail could be
    Could be one little...
    tiny universe.

  19. my answer is just... laughter.

  20. I'm not certain what the error in his reasoning is, but if you're rounding, anything less than a 0.5% chance is a 0% chance. :D

  21. An infinitely fine point cannot exist.

  22. I have not used probability or random variables in a long time and forget my calculus, but conceptually here it goes:

    The probability event is defined as "hitting" a position that is rangebound between 0 and 1 on a real number line. The real number line is continuous, as opposed to discrete- where the area between 0 and 1 would be undefined, so a 0% chance is not possible.

    Gene states that there is 0% chance of hitting a number but then says you'll hit some number, counterdicting his original statement.

    An infinitely small dart tip can be thought of as an impulse function or a delta dirac function, zero every except where it is, to which it integrates to 1.

  23. The problem with Callahan's argument is that his example proposes a case where the frequency distribution is asymptotic. That is, the probability of tossing any specific number on the number line would be extremely small, but it would not be zero. Hence, his example does not show what he thinks it shows.

    However, Callahan is right more generally to claim that a probability of zero does not mean that something is impossible. Probability is a measure of subjective belief, which means a probability of zero only means that some man (or men) believe it to be impossible. There is no such thing as an "objective" probability of zero.

  24. An infinitely fine-pointed dart has a tip diameter "approaching zero", since a non-zero-diameter tip is required in order to actually hit something.

    But then, in the same problem space the probability of hitting a real number with that dart is casually defined as "zero", when it should really be "approaching zero" since the tip diameter that would hit the number is itself "approaching zero" and not "zero".

  25. The initial claim is "X is impossible" and then you throw a dart. You will hit a point that is not X and therefore do nothing to disprove the initial claim.

    Precision is also an issue. Points along a real number line are uncountably infinite; whatever method is used to measure dart position will inevitably have limited accuracy. If you can only measure accurately to 10 digits then you introduce an error value of +/- 0.00000000001. This error range introduces a line segment with definite length which can be quantified in order to come up with the non-zero probability of the dart hitting a certain measurment.

  26. First, people say "0% chance" instead of "no chance" to sound scientific and reinforce the strength of their assessment. There assessment is wrong in most cases; few things that can be imagined are truly impossible.

    On the infinitely fine-pointed dart, it is presumably hitting an infinitely divisible number line, so if you said "0.4535", threw the dart and appeared to hit that point, you would find with your infinite measurement skills that it actually hit 0.4535000000175..., for example.

  27. > There's a 0% chance that X will happen."

    > Most people interpret that to be equivalent to, "X will never happen.

    This is true in finite realms. Dealing with mathematical infinity is when it gets weird.

    The probability of Gene's dart landing on any given number is 1/infinity. The limit of 1/x as x approaches infinity is 0. It seems sloppy to equate this limit with 0, but it may be mathematically accurate -- I'm not sure.

  28. An infinitely fine-pointed dart would not hit a number.

  29. It feels so obvious that there must be something I'm missing. But here goes:

    "there is a 0% chance you will hit it. But you *will* have hit *some* "

    Hitting "some" number is not the same as hitting a specific number. He's also saying there's no chance of hitting any number (presumably because the dart is so small) but then immediately says "you will have hit some number".

    Which makes no sense. His example is effectively, "there's a 0% chance of hitting a number, but of course we hit a number." Which doesn't seem to prove anything other than his premise is flawed.

  30. Mathematically and semantically I see no logical error (the concept is from real analysis or measure theory). But in the physical, measurable world, quantum mechanics does not let us see a pure point, rather only very small subintervals of the unit interval whose sizes are related to Planck's constant; the probablility or measure of such an interval is greater than zero. So Callahan's dart fails to get the point, so to speak.

  31. The best I can surmise is that he deduces that "zero percent probability" is as a result of an infinitely fine point tip when the probability is actually 1 over "infinity", not zero....which is different.

    Especially when dealing with "both" sides of an equation for example...because then you can't take things to "zero"-you can only cancel them.

    Did I just make sense or did I just make a fool out of myself?

  32. I'd guess it has something to do with interpreting words, beacuse it doesn't make much sense to me.

  33. It is impossible to have an infinitely fine pointed dart. Therefore such a dart could not possibly hit a number line. He is contradicting himself as well. He claims that the dart did in fact hit a number, yet he suggests that there was zero probability of it happening.

    This type of reasoning is why our country is in such bad shape.

    Remember, more than 44% of the Justices sitting on the U.S. Supreme Court in D.C. v. Heller interpreted the constitutional statement; "the right of the people to keep and bear arms shall not be infringed" to essentially mean "the right of the people to keep and bear arms does not exist."

    Anti-intellectualism has been standard practice for decades in academia and politics. It will be our demise.

    1. Gene just admitted in comments on his blog that if he told a person that there was 0% chance that X would happen, and then X happened a month later, he would argue to that person that his prediction was not necessarily wrong. Such a reasoning process is akin to Stephen Breyer in his dissent on D.C. v. Heller.

  34. 0% chance X will happen per definition means X will never happen.

    However, if his 0% chance is some sort of "rounding down" or the like, then of course he is cheating.

    And with the infinite fine-pointed dart, the problem is
    - such thing does not exist
    - in that case, the 0% comes as the limes of the function that assigns the probability of hitting number Y when we split the [0;1] into N disjunct and equally sized intervals, then we look at N->infinite.

    Since we cannot have infinite, we look where that function goes to.

    Problem is, it never reaches 0%, just gets infinitely close to it. ("for every epsilon radius environment of 0 there is a Z: f(x) <= 0 + |epsilon|, no matter how small we pick epsilon" yadda yadda)

    He just doesn't know what he talks about.

    But, as far as I know, this is secondary school maths. :)

  35. Callahan's error is that his example does not apply. Approaching 0 != 0. The difference between the two is the same as that between possible and impossible.

    "there is a 0% chance you will hit it"
    should be
    "the chance that you will hit it approaches 0"
    if he wanted his example to be correct.

    This is a Big difference. This means that the chance of hitting any number is non-zero (though only removed by an infinitesimal amount).

    Ergo, if one interprets a statement that the probability of the occurrence of some event is 0%, it would be correct to interpret that as "the event will never happen (is impossible)". Supposing that the issuer of said statement was accurate, of course...

  36. Probability is based on observations of randomly occurring events. The single throw of an infinitely fine-pointed dart at the entire real number line would still obey cause and effect – there must be a result.

    One may not be able to accurately say that with one throw of the dart you will hit any specific number, but you must hit a number. All his thought experiment shows is that you have a 100% probability of hitting some number that at the outset was mathematically impossible.

    I am not a logician or a mathematician, but wouldn’t assuming a 0% probability of hitting some number in the premises of an argument, then showing that there is a 100% probability of hitting some number in the conclusion be considered a probabilistic fallacy?

    - Jacob

  37. This sounds like a variation of the 'paradox of motion' that says you can never get from point A to point B because you can always travel half the distance between the remaining space.

    I'm hoping Wenzel drops some knowledge on us in an upcoming post.

    Here's my stab, he's started with a false premise that there is a 0% chance of hitting a specific number on the line but its really just infinitely small.

    That or it's a false dichotomy that excludes missing the line altogether. Which clearly happens all the time in risk management and economics.

  38. Probability is based on observations of randomly occurring events. The single throw of an infinitely fine-pointed dart at the entire real number line would still obey the rule of cause and effect – there must be a result.

    One may not be able to accurately say that with one throw of the dart you will hit any specific number, but you must hit a number. All his thought experiment shows is that you have a 100% probability of hitting some number that at the outset was mathematically impossible.

    I am not a logician or a mathematician, but wouldn’t assuming a 0% probability of hitting some number in the premises of an argument, then showing that there is a 100% probability of hitting some number in the conclusion be considered a probabilistic fallacy?

    - Jacob

  39. The target range is implied to he indiscrete and continuous in the setup of the question, but when he talks of a "number" being hit he must have discrerized the range at some granularity, otherwise how do you even express the number?

    For an expressible number to be hit the range must be discretized to at least the granularity of that number, which means the original statement of zero percent probability is false. Basically there is no possible value that you could substitute for X to make the statement true.

  40. His error is confusing limit of increasingly small intervals with a constant number (such as 0). And this has nothing to do with probability theory, it's basic calculus.

    What he is saying is that the derivative of any function is zero because the df is zero when dx is zero. Which is, of course, nonsense, because derivative is a limit of fraction Df/Dx, not a fraction itself (which is undefined when Dx is precisely zero) - note that I use capital D to denote non-infinitesimal delta to avoid confusion with common infinitesimal notation df and dx (which implicitly incorporates the limit operator).

  41. Umm...zero percent means zero percent...does it not?

  42. I think the theoretical math example just shows the absurdity of conceiving of reality as composed of infinitesimals. Reality is ultimately composed of discrete relationships.

    When people say "there's a 0 % chance that X will happen", they are referring to real events. Since Gene's example has a 0 % chance of being realized, it fails as a counter-example.

    (do I win ? :-)

  43. OK I'll bite.

    1. you can't have an infinitely pointed dart
    2. it's impossible to hit a line with dart because the line is one-dimensional and the dart is three dimensional so you would miss the line to the top or bottom.

    1. an infinitely pointed dart can never hit a finite number. The number it hit would by definition have an infinite number of digits, so the dart can't stop on the line because as soon as it hits a number, that number by definition has a finite number of digits.

  44. The size of "infinitely fine-pointed dart" is by definition zero, and thus cannot hit anything. It would be the slope leading to the point that actually hits.

    Thus the statement, "there is a 0% chance you will hit it. But you *will* have hit *some* number, so even though there was a zero percent probability that number would be hit, it was hit" contains one falsehood, and two sophistries.

    The falsehood is: "there is a 0% chance you will hit it."

    Of course, if YOU were going to hit it, then what does the dart have to do with anything? So I interpret this to mean, "there is a 0% chance you will hit it (with the dart)."

    This is false. The dart consists of more than a point.

    The first sophistry is, "But you *will* have hit *some* number,".

    This is true, but a sophistry. Why? It is a sophistry because it implies you will have hit *some* number WITH THE POINT. This isn't true.

    You can hit *some* number with the dart. But if the point is infinitely fine then it will simply pass through the target without collision. It would be the slope leading to the point that collides. If the point were not infinitely fine, then the probability of impact would not be zero.

    The SECOND SOPHISTRY IS, "so even though there was a zero percent probability that number would be hit, it was hit".

    If we can observe after the fact that it WAS HIT then by definition the probability was NOT ZERO.

    This is the problem with giving what Einstein called "denken experimente" primacy over observation.

    What a person THINKS about reality is an attempt to describe the objective object that reality entails. That the thinking involves a mathematical model is irrelevant.

    Thus, the objective observation must always have primacy. This is why it is IMPOSSIBLE to "prove" a theory. It is only possible to disprove one or fail (for now) to disprove one.

  45. There is an ambiguity in "There's a 0% probability that X will happen". If "X" means "that a number specified before the toss was hit", then the statement is true. If "X" means "that a number was hit" then the statement does not have a 0% probability. On the contrary, it is certain to occur on the conditions set forward in the example. No interpretation of "There's a 0% probability that X will happen" has been offered in which the statement is true and X happens.

  46. The probability would not be zero, it would be 1 divided by infinity.

  47. Well, he's right, for any given real number, the probability it is chosen is 0, but one will be chosen.

    The problem is that the act of choosing a real number isn't computable: It will take infinite time. For any probability distribution that is computable, It can output only among a computable and therefore countable set of choices. Each of these choices must therefore have non-zero probability.

    Here's another way to look at it, from information theory: The probability that an event occurs dictates the amount of information necessary to describe to someone else that it occurred. So choosing a particular 128 bit string at random has probability 1/2^128, and requires 128 bits to describe the event. An event with 0 probability would require infinite information to describe. This meshes with the description of choosing a real number above. Similarly an event that is absolutely certain requires 0 bits of information to describe that it occurred.

    1. Someone who actually understands mathematics has posted!

  48. With an infinitely fine pointed dart, theoretically, even if it hit the number line, one would never be able to determine what number it hit. There are an infinite number of real numbers between 0 and 1. The best one could do is determine that the dart "approached" a range of numbers, but upon further magnification that range of numbers it "approached" would appear smaller and smaller ad infinitum.

    Well, that is my attempt at the answer.

  49. If "X" was eventually hit, wouldn't that mean that the initial estimate that there is "0% chance" of it getting hit was wrong? The correct assumption would have been something to the effect of a number infinitely close to 0% that never actually reaches 0%.

  50. In a mathematical context, he is correct. But it is not applicable to any real world situation. He may as well have started his comment with, "Imagine a bingo tumbler with infinite pieces..." Thanks Gene, real deep thought.

  51. Gene Callahan's reasoning is an example of the No true Scotsman fallacy -- insofar as he reserves the right to infinitely add decimal places of precision to win the argument.

    1. JTG, you have confused experimental science with mathematics: the concept of "decimal places of precision" only applies in experimental science.

  52. It would be impossible to know, or even describe, what number the dart had hit. So even after the fact, there's no way you could make a statement like "the dart hit X".

    It would hit some trancendental number along the line of real numbers, which form an uncountably infinite set. And there's no way of describing them algebraicly (some like e or pi are described with symbols, but we still can't really specify the actual number precisely)

    So for the statement "There is 0% chance X will happen", I think it would be impossible to make that statement, even after the fact by looking at where the dart hit.

  53. I would side-step the dart example and go back to the original statement: "There's a 0% chance that X will happen."

    Due to the lack of precision in the measurement (i.e. only one digit) any actual value lower than 0.5% would round to the required 0%. For my own example, something that occurs 1/201 percent of the time is far from infinitesimal or impossible or "never". That's like saying April 4th NEVER occurs because 1/365 = 0% (when rounded to one digit).

    1. "Due to the lack of precision in the measurement (i.e. only one digit)..."

      Again, confusing math with experimental science.

  54. Brilliant! If something that can't physically exist encounters and object, it will not hit it. What happens to the rest of the dart, which probably has some mass and weight, when it encounters the dartboard, I wonder?

    Sounds like a sound basis for a Keynesian model to me. ;)

  55. These are fun academic games I suppose, but not to be taken too literally - reminds me of Zeno's dichotomy paradox ( http://en.wikipedia.org/wiki/Zeno%27s_paradoxes ).

    To me the apparent paradox is resolved by noting the fact that something of infinite smallness is, well, nothing. In other words if we throw "nothing", then it is clear that there is a 0% chance of hitting anything!

    Similarly, if the moving object in Zeno's dichotomy paradox is allowed to have any dimension at all (ie. not infinitely-small), the premise of the paradox breaks down.

    1. Liquefaction, you have kind of missed the idea of a thought experiment.

    2. Respectfully no - don't think I missed the point. Let me first say that I am happy you posted this as I think it makes for wonderful discussion. Love the EPJ, but I am a little puzzled by the tone of the discourse on this post. The Keynes reference(s) I read in the comments made me physically wince away from the screen.

      I understand that any finite number divided by infinity is taken as 0, which this thought experiment is riffing on. There is nothing controversial about that. "Standard" as you mention. I do think there is a bit of hand waving in the problem statement, though. Let me distill the experiment that you have relayed without the window dressing to point out the triviality:

      "Pick any real between 0 and 1 (I will wait...). Oh you picked 0.618, what a fantastic choice, but I hope you are sitting down for this. It does not exist because it has an infinitely small chance of existing, say 1 divided by all possible reals except for itself, which is still 1/(infinity-1)= 0."

      To me this is more of a cause for reflection on the formulation than some sort of deeper insight into the nature of probability.

    3. "I understand that any finite number divided by infinity is taken as 0.."

      That is incorrect: it is undefined.

      "It does not exist because it has an infinitely small chance of existing..."

      No, this is all wrong. See the Wikipedia article on continuous distribution probability.

    4. "that is incorrect: it is undefined."

      I agree with that but your problem statement did not say
      "...for any particular number in that range, there is an UNDEFINED chance you will hit it." Your problem statement said 0%.

      You also refer to a nifty proof posted on your blog, not disagreeing with it at all - but doesn't it prove by contradiction that the probability of hitting any real must be 0? Isn't that the whole point of the proof? Isn't this another way of saying that 1/OO = 0? What am I missing here?

      Let help spell out the correlation of my example to yours.

      "toss an infinitely fine-pointed dart at the real number line and hit a spot between 0 and 1"

      "for any particular number in that range, there is a 0% chance you will hit it."

      "But you *will* have hit *some* number"

      The same cruddy "paradox", if stated a little differently. not that anyone is reading the comments anymore but had to go on the record...

  56. Absolutely incredible, folks! I did not invent this example: it is absolutely standard probability theory. Go ask any math PhD.

    I think next I will post, "The square root of two is an irrational number," and then wait for the stampede of lemmings to arrive here declaring "Callahan has just shown his Keynesian irrationality."

    1. That would not be a very interesting post Gene.

  57. Actually, Gene, if you ask any math Ph.D., they will tell you that for continuous distributions P(X=x) isn't defined. So your original claim that the probability of selecting any real number on the interval [0,1] is equal to zero is actually a meaningless statement.

    1. David G. are you sure about that? For the non-important point: I think what you mean to say is that Gene's statement was false, not that it was meaningless. If I claim, "The largest number in the range [0, 1) is 1," I didn't say something meaningless, I said something false. (Note that in the example, I am trying to assign a value to an undefined concept, namely the largest number in the range [0, 1).)

      But beyond that, let's make sure I understand what you are claiming.

      ==> We agree that Pr (0 We agree that Pr (0 < x < 1/2) = 1/2, right?

      ==> We agree that Pr (0 < x < 1/4) = 1/4, right?

      ==> But you're saying that Pr (x=0) is undefined? You don't think it's well-defined, and is 0?

    2. @David G: "Actually, Gene, if you ask any math Ph.D., they will tell you that for continuous distributions P(X=x) isn't defined."

      That's funny, David G, because I GOT the dart example from my boss, a math PhD and the author of two books on probability theory, Randolph Nelson. Clay Shonkwiller, a math PhD, came to my original post and called this a good example. I was alerted to your post by a reader highly educated in math who said "David G" is bluffing.

      So, David G, where is your math PhD from? And your full name?

      And while your at it, you should edit the Wikipedia entry on this -- let's see how long your edit lasts:

      "While for a discrete distribution an event with probability zero is impossible (e.g. rolling 3½ on a standard die is impossible, and has probability zero), this is not so in the case of a continuous random variable. For example, if one measures the width of an oak leaf, the result of 3½ cm is possible, however it has probability zero because there are uncountably many other potential values even between 3 cm and 4 cm. Each of these individual outcomes has probability zero, yet the probability that the outcome will fall into the interval (3 cm, 4 cm) is nonzero."

    3. Bob, my point is that Gene's statement is meaningless. In the same way, "gobbledegook fluoresces" is meaningless rather than false because "gobbledegook" is a meaningless word.

      Yes, I am saying that P(X=0) is not well-defined for a continuous distribution and so is not 0.

      Gene, if you are so sure that my statement was false and have access to great experts in the field of probability, then by all means provide a nontrivial definition of P(X=x) for continuous distributions (by nontrivial, I mean that you don't simply declare P(X=x)=0, but that you can derive that conclusion from the definition). You can muster whatever authority you wish, but if you can't give a definition, then my point stands. If it turns out that there is such a definition, then I will stand corrected, but until you produce one, I would ask that you be just a bit more courteous in your tone.

    4. Well, one obvious proof is that the integral from x to x is 0 in either the Lebesgue or Riemann integral. Getting deeper into this problem requires understanding how probability spaces are defined and why, which requires a bit more thought.
      It's also strange to declare that something is not well-defined when one does not like the definition.

    5. Gene, the wikipedia article you cited contains a further passage:

      "This apparent paradox is resolved by the fact that the probability that X attains some value within an infinite set, such as an interval, cannot be found by naively adding the probabilities for individual values. Formally, each value has an infinitesimally small probability, which statistically is equivalent to zero."

      The point of dispute resolves around the issue of whether or not an infinitesimally small positive number can be considered as equal to zero.

      You and your boss say yes, David G and Wenzel say no.

      In my view, an infinitesimally small number is not well defined. It cannot be "grasped" as a definite object that has an actual value.

      Most mathematicians equate 0.99999.... with 1.

      I do not.


      Because of this reasoning:

      0.9 < 1

      0.99 < 1

      0.999 < 1

      0.9999 < 1

      0.99999 < 1

      0.999999 < 1

      0.9999999 < 1

      0.99999999 < 1

      0.999999999 < 1

      0.9999999999 < 1


      We can continue to add an additional 9 indefinitely, and at no point is there any logic that would require us to change the "<" to a "=".

    6. David G. wrote: Gene, if you are so sure that my statement was false and have access to great experts in the field of probability, then by all means provide a nontrivial definition of P(X=x) for continuous distributions (by nontrivial, I mean that you don't simply declare P(X=x)=0, but that you can derive that conclusion from the definition).

      OK it's late and I am not a math PhD, but David G. I think it would go something like this:

      ==> Because the dart isn't biased, assume a uniform probability density function. Since the interval is [0,1] that means pdf(x)=1 for 0<=x<=1.

      ==> Then define P(a<=X<=b) = a-b, for 0<=a,b<=1.

      With this definition, obvious probabilities work out. The probability that X will fall between 0 and 1 is (1 - 0) = 1. The probability that X will fall between 0 and 1/2 is (1/2-0) = 1/2.

      So then if you ask, what's the probability that X is *exactly* 3/8 (say), then you write:

      Pr(3/8<=X<=3/8) = (3/8 - 3/8) = 0.

      I would be stunned if you could produce for us a math PhD who said this type of approach is nonsense, and in fact the probability in question is undefined.

    7. Oops David G. I can't see my comment now (still awaiting moderation), but I think I skipped a step: You *define* the probability that X will fall in between a and b, by integrating the probability density function over x, and evaluating at those end points. So the solution to that definition gives you the formula of (b-a).

    8. OK, David G, you said that any math PhD would agree with you. Now, after being pointed to several who don't, you completely change tack and ask me for a definition that yields this result. David, I am NOT a PhD mathematician. In a field in which I am not an expert, I take the word of experts. Apparently, you found it discourteous for me to ask where your PhD is from, and what your full name is. Unfortunately, I am going to be discourteous again: where is your PhD from, and what is your full name?

    9. Bob, I'm afraid your attempted definition doesn't quite work. The problem is that it produces P(X=x)=0 for discrete distributions as well. A better definition would be the limit as h goes to zero of F(x+h)-F(x-h), where F is the cumulative distribution function. This, however, runs into the problem of boundary points, so we could define a one-sided limit for those cases. However, my point is not that you couldn't cook up some definition for P(X=x), but rather that you would have to cook one up because there isn't a preexisting definition.

      Gene, you seem to have misunderstood what I've written. I have not changed tack at all. My original claim had two points: (1) that P(X=x) is not defined for continuous distributions, and (2) that anyone with a Ph.D. in mathematics would agree. What you seem to have missed is that if (1) is true, then any math Ph.D. you proffer as a counterexample to (2) doesn't know what he's talking about, at least on this one issue. I have a sneaking suspicion that if you were to press your friends for a definition of P(X=x) for continuous distributions they would admit that one doesn't exist and so they'd have to cook one up. Notice here, as in my last post, I'm not asking you to produce the definition yourself: ask you friends, the great experts. And if your friends have a preexisting definition, then please, for the sake of our mutual enlightenment, share it! If you can do so, I will withdraw my complaint and even apologize for my earlier impudence. Surely there is no need to be vindictive with these generous terms on the table?

    10. David G, you are simply wrong. For one, Bob's example cumulative distribution only applies to continuous distributions, for obvious reasons. Saying that it fails because it does not apply to discrete distributions is nonsense. He is giving a special case where the result is true because you refuse to take the definition given on Wikipedia and in all probability textbooks I have ever seen seriously.

      Probability is clearly defined on single points because probability is defined over a certain complicated collection of events, which includes single events. This is true by definition.

      There are many ways of showing that P(X=x)=0, but apparently, none of them are good enough for you.

      Here is the easy version:
      Integrals from a number to itself are always 0.
      Here is the complicated version:
      Any probability is absolutely continuous with respect to the normal measure, M, for which M(x)=0 for a single x, and therefore single points are of 0 measure/probability.

      Note that neither of these make use of limits or infinity or any other “controversial” idea.

    11. ziragt, I'm afraid you're somewhat mistaken about what I've written. As I wrote to Bob, I am not saying that it is impossible to come up with some definition of P(X=x) for continuous distributions, but merely that it doesn't have a preexisting definition. Nor am I disputing that if it were to be defined, that it would be 0--in fact, the definition that I gave to Bob as a correction of his own attempt produces just that. I was not aware of that measure-theoretical definition; can you provide some kind of academic documentation to show that it is actually the standard definition used in by mathematicians? If you do not have it, perhaps Gene's friends can verify it.

    12. David, I am not coming up with an ad hoc definition. I am trying to state that P(X=x) is defined, using completely standard concepts. I am not sure what else there is to say about this.

      As for a reference, I suppose that Kai-Lai Chung's textbook, "A Course in Probability Theory", would state it, although I don't have my copy on hand to verify this. Most graduate textbooks should say something similar.

      Anyway, thanks for motivating me to relearn some probability theory.

    13. ziragt, you're welcome. I guess this just goes to show that even when both sides are confident in their positions, this sort of discussion can still be fruitful for them both.

      Given your source, I will gladly accept that the measure-theoretical definition you've given is widely used--though if you could check the actual text, that would be doubly reassuring. I had hoped to conclude my dialogue with Gene in a manner perhaps like this, but I can't really blame him for having better things to do than argue with me.

      Thank you for pursuing this discussion with me.

  58. Does Callahan have a job, or does he sit in a library all day long?

    1. Richie asked, "Does Callahan have a job, or does he sit in a library all day long?"

      What if Callahan is a librarian in a wheelchair?

    2. Thanks for asking, Richie! I am posting today from the British POlitical Studies Association conference in Belfast, where I presented my paper on Bishop Berkeley and have been pitching my fourth book. But after this, it's back to teaching at Purchase College, editing, and getting cracking on that fourth book!

  59. What's amazing to me is that with all the incredibly important posts about monetary policy, Fed abuses, civil rights abuses, etc. on this blog, the blog entry that gets the record number of replies is some asinine argument over infinitesimals (literally). In the end, you may come up with some answer that makes equations work, but nobody can tell you what it really means -- like "photons have zero rest mass" (a result that comes from a requirement derived from the Lorentzian mass equation for the speed of light).

    That reminds me about arguments about the right to keep and bear arms where there's always some "genius" who says something on the order of "what about the guy who owns a tritium-based implosion device triggered by quantum-entangled photons to create a black hold and suck the whole universe into nothingness? Does he have a right to do THAT?" As if physics, economics, and basic humanity can't stop a guy from creating the ultimate doomsday device, but a bunch of assholes in Washington D.C. can solve the problem by passing a law and hiring a bunch of bureaucrats.

    But the real answer is that when most people say "X will never happen," they mean "there is an infinitesimally small chance that X will happen." We use the first phrase to avoid being lumped into the group of people who never get laid.

    And there are more important things to worry about.

  60. If it is of any interest, I am not the "David G." who has posted above.

    1. David Gordon wrote: If it is of any interest, I am not the "David G." who has posted above.

      I agree that there is a 0% probability that the careful academic I know would, with such confidence, utter demonstrably false things. However, I won't go so far as to say it's literally impossible that you two are the same guy.