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 lines 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. :)

However, I found most interesting the comment made by David Gordon. He took a unique view on Callahan's statement, not made by any of the other 50 plus commenters.

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.David, in other words, blows Callahan's comment out of the water right at the starting gate. They don't call David a genius without good reason.

BTW, David was good friends with both Murray Rothbard and Robert Nozick. Their phone conversations must have been epic. Maybe we can someday coax the NSA to release the recordings.

Nice one David.

ReplyDeleteYou got to the *point* much more quickly than I.

Sounds like Scholasticism pure and simple to me! Angels sitting on a pin head, etc., etc.

ReplyDeleteAquinas and Scotus would be proud.

Scholasticism is one of the great intellectual achievements in the west, and as Rothbard argued was an essential underpinning to the natural rights theories that gave rise to the cultural acceptance of capitalism, and of the acquisition and investment of wealth. You might consider reading something about it beyond pulp fiction.

DeleteI'm afraid that I don't understand what is so insightful about Gordon's posting.

ReplyDeleteHis first interpretation of the Callahan's statement(i.e., "X" means "that a number specified before the toss was hit", then the statement is true") is saying that the probability after the fact is 0. What's so insightful about that? Saying that probability is zero AFTER the experiment has occurred is actually a misleading use of the term "probability." There is no such thing as uncertainty or probability if we know the outcome, as Ludwig von Mises himself said!

His second interpretation of the statement (i.e., "If "X" means "that a number was hit" then the statement does not have a 0% probability") is exactly what all of the other posters said! namely, he is merely saying that Callahan is wrong to think that there is a 0% probability ex ante. There's nothing more insightful in that than what everyone else wrote!

Can someone please explain to me what is especially insightful about that?

What I gather he is saying is that the description is not actually any sort of paradox, it's just two separate statements that aren't actually related to one another. If you choose a number and then throw your dart at the number line it will not land on the number you picked, ever. There is a zero percent chance it will happen. The fact that it lands on a number is irrelevant since that number will not be the one you selected.

DeleteI intended my first interpretation to be a probability before the experiment, not after it.

DeleteDan, are you seriously suggesting that it is literally impossible to hit the number you chose when you toss the dart? If so, then how can that outcome be considered part of the sample space, and thus be part of the probability calculus at all?

DeleteThe same question applies to Dr. Gordon's comment here. How can we say that Callahan's statement was true (as Dr. Gordon says here) unless we assume that it is literally impossible to hit the number before the experiment? Utilizing the classical approach, we could only make this assumption if we exclude the outcome from the sample space ex ante altogether.

The problem here is that everyone is simply accepting the application of an a priori "classical" method for generating a probability when it is inappropriate in this case. The classical method was developed for gambling situations involving k discrete outcomes, all of which are assumed to have an equal likelihood of occurrence. The method, in other words can't be legitimately applied to continuous variables, as marketsclear correctly notes below.

More importantly, everyone seems to uncritically accept that there is an "objective" probability of zero of tossing any given number, but this is highly debatable, to say the least. This would only be possible if we assume that the world is indeterministic and random. It is NOT possible for there to exist "objective" probabilities at all in a deterministic world. The great probabilist I.J. Good made this important point.

Here's my take on this (I'm a graduate student of mathematics):

ReplyDeleteAs a Rothbardian, I have my disagreements with Gene Callahan, but in this case, he is 100% correct. He's simply stating a very basic fact about probability theory.

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

There is no such definition.

"And with the infinite fine-pointed dart, the problem is

- such thing does not exist"

Yes, of course we cannot literally throw a dart at the real number line, since the real number line isn't a physical object. Gene is describing a thought experiment where a real number between 0 and 1 is randomly chosen. Then the following statements are true:

1) The probability that the number 0.5 will be chosen is 0.

2) It is possible that the number 0.5 will be chosen.

(For a proof, see here.)

The limit of 1/n as n approaches infinity is 0, not "infinitely close to it." (The limit of a convergent sequence of real numbers must be a specific real number.) But you don't even need to use limits to show that the probability is zero (see the link above).

David Gordon is a fine philosopher, but I found his comment incomprehensible. Perhaps he's trying to point out that from the perspective of someone who already knows that the number 0.5 was chosen, the probability that 0.5 was chosen is 1, not 0. This is true, but from the perspective of someone who is going to conduct Gene's thought experiment, the statements 1) and 2) are true.

But the probability in 1 is not exactly zero. It approaches zero as the fineness of the measurement increases. But there is no perfectly fine measurement in real life, no matter how close we come in theory. So there is always a positive nonzero probability, no matter how close to zero it is.

DeleteThere are ~7 billion people in the world. The chances of anyone being the best at anything in any given year are practically zero, but every year, at every thing, SOMEONE is best, and rarely got that way by accident.

Thus, the premise seems to discount free choice having any effect on the measured outcome, even when surveying individual preferences.

My objection to Gene Callahan's post isn't based on the mathematics of probability but rather on the meaning of the sentence he wants to challenge. Gene wants a counterexample to "0% chance X will happen per definition means X will never happen." Suppose someone predicts that, when Gene's dart is tossed, it will land on the number 0.5 The prediction has a 0% chance of happening. If the dart is then tossed and it lands on 0.5, then this is a counterexample to the statement that Gene challenges.

DeleteBut in the example Gene discusses, the dart is tossed and lands on a number---let's assume, once more, that it's 0.5. Gene wants to say, "The dart had a 0% probability of landing on 0.5, but nevertheless it did land on 0.5. Thus it's false that '0% chance X will happen per definition means X will never happen.'"

I don't think the example shows this. What the example shows is that if someone were to have predicted that the dart would land on 0.5, his prediction would have had a 0% chance of being true. But that is a counterfactual--"if someone were to have made a prediction, then such-and-such would follow." But to be a valid counterexample to "0% chance X will happen per definition means X will never happen", Gene needs not a counterfactual but an indicative statement: Someone predicts X will happen, where X has a 0% chance of happening, and X happens.

David K,

DeleteI have a problem with point (2) of your statement. Is it really possible that 0.5 will be chosen? it seems to me that the dart will be guaranteed to hit a transcendental number, which form an uncountable set along the real number line. Since there are infinitely more non-algebraic numbers along the line, you'll be guaranteed to hit one of them (ie. you'll be guaranteed not to hit an algebraic number).

Since transcendental numbers cannot be described algebraicly, or with perfect precision, it's impossible to say "the dart hit X", even after the fact.

Thus for any value "X" that is specified in the statement "the dart will hit X", the dart will never hit that value, even if thrown an infinite number of times.

Or another way of stating it: for any event with non-discrete outcomes (ie. with continuously varying outcomes) you can never know exactly what happened, even after the fact, because you can only know the result within a certain precision. So it's impossible to say "X happened", because "X" is not knowable. So if you say beforehand "X will never happen", that is a true statement for any "X" that can be stated.

I'm a big fan of the blog, but I have to say that Richard and Gene are perfectly right here.

DeleteAn event occurring with probability zero would never happen if we are dealing with a discrete distribution. However, Gene's thought experiment involves a continuous distribution, in which we general say an event with probability zero happens "almost never".

The probability of zero occurs through a limiting process, usually a Lebesgue integral on a suitable probability space. However, the probability itself is defined to be the limit of this process or the evaluation of the integral to be exact. While the integral represents a limit of increasingly good approximations, the probability itself means the result of this. The probability itself is not an estimate.

The confusion comes from dealing with uncountable sets and the strict definition of probability. The probability of you hitting any given number is zero, but it can happen. The probability of you hitting an irrational number is one, but it is possible for that to not happen.

My probability math may be rusty.... but here is my sense of this.

DeleteInfinity in a set of points on a real curve is just a non-real abstraction. In reality if we imagine a real dart, and a real set of points in a possible space that it can strike, then the larger the number of points (and the finer the point of the dart) then the smaller the probability of hitting any designated point. As the number of points approaches infinity the probability of hitting a given point approaches zero.

Even if we imagine a truly infinite set of points (an unrealistic example now), still, I don't think the probability of hitting a given point is literally zero. It seems to me that 1/n if n is infinity is not 0, any more than 1/0 = infinity. 1/0 is undefined, not infinity. Likewise, since there is no number "infinity," 1/infinity ought to in a sense also be undefined--not zero. We can say that 1/n as n goes to zero, approaches infinity. And 1/n, as n goes to infinity, approaches zero. I would think that 1/infinity = infinitesimal, not zero.

So the chance of hitting any given point of an infinite number of points n is not 0%. It is an infinitely small (infinitesimal) chance, but not zero.

Or so it seems to me.

OK, riddle me this: Does an an infinitely deep well have a bottom?

ReplyDeleteDavid,

ReplyDeleteOne thing you neglect to note is that both you and Callahan are calculating the probability for the event occurring a priori using the so-called "classical method" by assuming each outcome has equal likelihood of occurring. Frequentists like Richard von Mises (and many Austrians, unfortunately), however claim that there can be no such thing as a probability until the dart has been tossed many, many times. Only then can a relative frequency of occurrence be calculated, which frequentists interpret as the ONLY "real" probability.

Hence, the thought experiment itself forces us to accept the classical method for calculating probabilities, which many Austrians are loathe to accept. This just adds another level of complexity and absurdity to this thought experiment.

I, on the other hand, define probability subjectively (http://libertarianpapers.org/2009/26-crovelli-on-the-possibility-of-assigning-probabilities-to-singular-cases/), which means probability is merely a measure of subjective belief. Armed with a subjective definition, we don't have to resort to ridiculous thought experiments to point out that something with a 0% probability can still occur. All a 0% probability means is that some person THINKS it is impossible, not that it is IN FACT impossible.

...isn't there a difference between 'approaches' and 'reaches'? The limit of 1/n as n approaches infinity - a concept, let us remember, not a number - is 0, as you say, but I was under the impression that the whole point of a limit is that it isn't ever *actually* reached, just approached. In practical use, it may be fine not to make a distinction between 'infinitesimally close to X' and 'X', but as a matter of philosophy, surely the distinction is real?

ReplyDeleteOne other note about the classical method Callahan is using. The classical method STARTS from the assumption that each POSSIBLE outcome in the sample space has an equal likelihood of occurrence. In other words, the method itself takes as given that each outcome IS possible. Callahan could have saved his breath by just pointing this out (i.e., that every outcome IS possible according to the classical method itself) instead of dragging the whole discussion into a pointless discussion of asymptotes and the limit.

ReplyDeleteWould NSA bureau(c)rats even know what you are talking about? Liberty? Private Property? Self-responsibility? WHATZ THAT???

ReplyDeleteEven if a few NSA drones could be found who wanted to co-operate with you to uncover and publish recordings of conversations of David Gordon with Robert Nozick and Murray Rothbard, I doubt very much if they would know how to access the relevant files.

PEACE AND FREEDOM

What exactly is the relevance of all this to economics or political philosophy?

ReplyDeleteIt's the study of logic. Which underpins economics and political philosophy. You won't get to correct answers in advanced sciences like economics or political philosophy if you don't master the basics.

DeleteIf I understand David correctly, I agree with him. Actually, it was the first thing I thought when I read the post. I think what David is saying, put more simply, is this: If you're throwing the infinitely-tiny-pointed dart with the INTENTION of hitting the number 0.23929398492839489283984982983949384982, then there is a 0% probability that you will succeed. But, to the contrary, if the goal is to hit ANY number (as the condition of 'x' is defined in the example) then you're going to succeed no matter what.

ReplyDeleteDavid K,

ReplyDeleteAs a math student, you should know that to talk about the probability of hitting a point in a continuous distribution is nonsensical. Gordon is saying that if it is possible for the dart to hit a number, then the probability of that event cannot be zero. If the probability of hitting a specific number is exactly zero, then it must also be true that the probability of hitting several numbers is also zero, since the probability of the union of events is less then or equal to the sum. However, once we make a range of points, it is clear the probability is non-zero. If we sum exactly zero an infinite number of times, the result is still zero; however the integral over an area of points yields a non-zero result.

Probability is firstly a logical science dealing with determining the likelihood of events. If mathematical tricks yield a result that is incongruent with logic, then you're doing something wrong. In this case the limit for any point is zero, yet the integral over a range of points is non-zero. The attempt to discretize a continuous distribution in such a manner is invalid.

You're trying to apply reasoning of discrete events to your analysis of the continuous. You're making the ill fated logical jump from the finite to the infinite that led to erroneous mathematics for so many years.

DeleteThese are not mathematical "tricks". They're perfectly valid techniques that allow for a systematic study of probability. This is not leading to something counterintuitive or illogical. You simply a priori have decided that a probability of zero must mean impossibility. We define probability as the outcome of a process which may result in zero. You have attached your own extraneous meanings to the number. That does not make the science invalid.

What interests me most of all about this discussion is that no Austrians are screaming and moaning about the method for generating probability that Callahan uses. Austrians of the Mises-Rothbard bent have followed Richard von Mises in claiming that the a posteriori relative frequency method is the ONLY method for generating legitimate numerical probabilities. They have, moreover, argued that probability cannot legitimately be applied to single cases, like the one Callahan proposes.

ReplyDeleteHow come no one is whining about Callahan's a priori approach to generating numerical probabilities for a single case? Why is no one rushing to Rothbard and Mises's approach to probability? Have the Austrians finally abandonded Rothbard and Mises's ridiculous theory of probability? Are we all subjectivists now?

I certainly hope so!

Every paradox is the result of at least one false premise -- always. So why the discussion.

ReplyDeleteBTW: The false premise is the infinitely fine pointed dart. Go find me one.

Jim, I'm surprised that you of all people are not jumping to the defense of the Mises-Rothbard theory of probability here. Why is it suddenly possible to calculate a probability a priori in this example? There is no "class" involved, as Ludwig von Mises puts it, so why is it suddenly ok to assign a numerical probability in this case?

ReplyDeleteI'm just curious.

What's the probability that anyone except Murphy and Gordon actually read Callahan's droolings?

ReplyDeleteMark,

ReplyDeleteHaven't we been through this before? Mises wrote a book, not a passage. If you read beyond class probability, you will encounter case probability. And, yes, numerical probabilites can be assigned, but they are metaphor ...I'll stop there since I don't want to spoil the ending for you.