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Some common misconceptions about probability

I have been writing about probability in the Austrian literature for three years (see here, here, here, here, here, here and here).  While I am quite sure that many people are tiring of my articles at this point, I feel that it is important to highlight some of the more obvious misconceptions about probability that I continue to confront whenever I discuss probability with Austrians.  I do this in the hope that we can stop getting tripped up and distracted by these errors and finally have a serious discussion about the topic. 

Misconception #1: Casino's Don't Predict the Outcome of Sporting Events

The most common misconception about probability that I encounter stems from an example that I used in an article I wrote several years ago.  In the article, I claimed that since casinos and bookies generate numerical probabilities for singular sporting events (like boxing matches), this is strong prima fascia evidence that Austrians have been wrong to claim that numerical probabilities cannot be assigned to singular events.  If I could go back in time and use a different example in the paper (e.g., the fact that geologists and probabilists generate numerical probabilities that oil will be found in singular oil plays), I most certainly would, because the example has unintentionally caused an immense amount of confusion.  

This example has caused so much confusion, in the first place, because many people do not fully understand what casinos do when they take sports bets.  They think that no prediction is involved in generating sports odds because casinos don't care who wins the game.  All the casinos hope to do is to make money from the commissions on the bets.  Thus, according to this common line of thinking, no prediction is involved, because the casino simply needs to figure out the odds that will "balance the books" by equalizing the bets for and against.  (Bob Wenzel recently made this naïve mistake in a conversation with me, for example).    

The obvious problem with this line of thinking, however, is the fact that casinos have to predict the odds that will balance the books!  Casinos don't have a crystal ball to consult in order to know what the odds "should" be on any given sporting event in order to induce the same amount of money to be betted for and against a team.  They have to predict the odds that will balance the books.  In the case of the Super Bowl, for example, which can involve millions of dollars being wagered on either side of the line, casinos are not in a position to know a priori how the thousands upon thousands of people around the world intend to bet on the game.  They have to predict how they think people will wager.  And how do casinos predict something as ethereal as the way tens of thousands of people that they don't even know might bet on a single game?  They predict the outcome of the game of course! 

Casinos obviously modify their prediction of the winner in the light of other factors that are relevant for taking bets (like the sizes of the cities where bets are expected to come from), and they obviously adjust the odds as time passes in order to try to balance the books if their initial odds fail to do so, but the most important thing for a casino to predict is the winner, because this prediction forms the core of the model that is used to predict how the bets will come in.  If the model as a whole correctly predicts how people will bet, the books "balance" and the casino makes money, but if the model gets it wrong, as many casinos came very close to doing in the last Super Bowl, the casino loses money.  In fact, the casino can lose massive amounts of money if its model fails to accurately predict how people will bet.

To put this matter to rest for once and for all, therefore, the generation of sports odds in a casino obviously involves predicting the outcome of the game in order to predict how bets will be placed.  The outcome of the game is not the primary interest to the casino, but it is nevertheless critical for generating odds that will balance the books.

Whether or not we want to call the sports odds generated by casinos "probabilities" is different question that must be addressed, however.

Misconception #2: Sports Odds are Not Probabilities Because They are Not Numbers Between Zero and One

As I just noted, my choice to include an example drawn from gambling in my paper turned out to be a mistake, because people have a lot of misconceptions about gambling odds in general and sports odds in particular.  It would have been far better for me to have used a different example to illustrate my point that Austrians are wrong to claim that probabilities cannot be assigned to singular events. A better example might have been the fact that weather forecasters assign numerical probabilities that it will rain to specific days (e.g., 40% chance of rain on Wednesday), and not to a "class" of days, whatever that might mean.

Another reason why my gambling example has caused confusion stems from the fact that many people do not think sports odds are probabilities.  They think sports odds are not probabilities either because 1) sports odds are not expressed as numbers between 0 and 1, or because 2) sports odds do not involve relative frequencies of occurrence.  Both of these objections to calling sports odds "probabilities" fail miserably, however.

In the first place, the fact that sports odds are not expressed as numbers between 0 and 1 is completely beside the point, because they can be converted into numbers between 0 and 1 at the drop of a hat.  There is a ridiculously simple procedure for doing this.  There is no mathematical or philosophical reason why gamblers need to say "5 to 1 odds" instead of saying ".833."  The fact that gamblers prefer to say "5 to 1 odds" is simply due to the fact that gamblers and casinos historically adopted the usage, not because they are mathematically different from one another.  Since sports odds can be easily converted into numbers between 0 and 1, it is beyond silly to object to calling them "probabilities" until they are actually converted.  This is like saying that the word "perro" does not mean "dog" until it is converted into English, which is beyond silly.  Just convert the damn odds into numbers between 0 and 1 if you care so much and get over it!

Misconception #3: Sports Odds are Not Probabilities Because They are Not Relative Frequencies

The objection that I more commonly encounter from Austrians is not as silly as objecting to the form in which sports odds are expressed.  Austrians have more commonly objected to calling sports odds probabilities because Ludwig von Mises and his brother Richard von Mises said so.  And why did the brothers von Mises say that sports odds are not probabilities?  Because, according to the brothers von Mises, numerical probabilities can only be assigned to "classes" or "collectives" of events for which long-run relative frequencies can be calculated.  Assigning numerical probabilities to singular events like sporting events is nonsensical, they claimed, because no relative frequencies can conceivably be calculated.

Since this claim apparently still rings true for a number of Austrians (like Daniel Sanchez and Hans-Hermann Hoppe, for example), it is important to point out how baldly question begging it is in this context.  If we are trying to figure out whether sports odds "are" probabilities, it is question begging to say that they are not probabilities simply because they are not derived from long-run relative frequencies.  To assume from the very beginning that only relative frequencies are "real" probabilities is to assume the very thing one is attempting to prove.  It is true that the relative frequency method can only be utilized in cases where a "class" or a "collective" of similar events can be mentally constructed, but to argue that this fact constitutes a justification for saying that only relative frequencies "are" probabilities is a classic case of circular reasoning.  

The truth of this can be clearly seen if we consider what would have happened if the classicists had utilized this same type of argument.  If the classicists had claimed that the only "true" or "real" probabilities are those that are generated with the classical method by assuming equal likelihood of occurrence (a claim the classicists were more justified to make than the frequentists, since they were the first ones to systematically generate numerical probabilities), frequentists like the von Mises brothers would have cried foul.  The frequentists would have justifiably said "Look, you classicists are assuming the very thing you are attempting to prove!  You are just defining probability as your favorite method, and you are simply defining our method out of existence.  That's not fair."  The end result would have been two schools of thought defining each other's methods out of existence by defining probability in terms of their own favorite methods.  

Whether on not sports odds, relative frequencies, or any other numbers "are" probabilities depends upon the definition for probability that we adopt.  We need to know what probability is before we can say whether any particular numbers "are" probabilities.  Equally important, we need to know what probability is before we can say that any particular method is capable of generating numerical probabilities.  To assume from the outset that the relative frequency method (or any other method, for that matter) is a legitimate method for generating probabilities is to assume the very thing one is attempting to prove.   

Misconception #4: Only One Method is Capable of Generating Numerical Probabilities

Prior to Richard von Mises (and his intellectual forbearer, John Venn), the few people who talked about numerical probability viewed it as a simple matter of looking at the possible outcomes in any given situation.  In the case of flipping a coin, for example, the classicists said that there are two possible outcomes, heads and tails, and they simply assumed that each possible outcome has an equal likelihood of occurring.  The probability of tails occurring if we flip a coin would thus be ½.    

Richard von Mises's response to this "classical" approach to probability was simply to define it out of existence.  He simply defined probability as a long-run relative frequency of occurrence and called every other method for generating probabilities "absurd" and "meaningless"?including the classical method.  Talk about revisionist and supercilious ingratitude!  Richard von Mises simply defined out of existence the very method that gave birth to probability in the first place!

For Austrians, Richard von Mises's definition for probability ought to raise a number of red flags.  In the first place, even if we agreed with Richard von Mises that probability is a "real" physical property of things in the world, (a claim that is extremely dubious), what justification is there for saying that the relative frequency method is the only method that is capable of measuring it?  Is it not the height of arrogance and dogmatism to say a priori that only one method will ever be capable of measuring this property?  This is very much like saying that only the imperial measurement system will ever be capable of measuring length or weight, which is completely absurd. 

At the very least, this claim ought to arouse our suspicion that other methods might be available to estimate numerical probabilities, even in those people who might be priggish about labeling them "real" probabilities.  The classical method just discussed, for example, is clearly a decent estimator of the relative frequency at which certain events will occur.  If I pick up a die off the table, and neither one of us knows whether or not it is truly fair, surely the classical method for determining probabilities gives us a decent starting estimate of the likelihood of tossing a two with it, even if it turns out to be slightly off a posteriori.  If I utilize the classical method and say that the probability of tossing a two is 1/6, is that really a "nonsensical" thing to say?  What about other methods we might utilize, like Monte Carlo simulations?  These methods are all absurd and meaningless, simply because they are not derived from long-run relative frequencies? 

This question is all the more apposite here, because frequentists like Richard von Mises are not actually able in practice to obtain the "limiting value" of the relative frequency of occurrence for a class of events.  If I toss a coin a million times, this might get me closer to the limiting value of the relative frequency of occurrence of tails than if I only tossed it one thousand times, but one would literally have to toss the coin infinitely to obtain the "real" limiting value (Richard von Mises insists on this point).  Since it is completely impossible for human beings to perform actions infinitely or indefinitely, however, this means that all relative frequencies that we calculate in the real world are nothing but estimates of the limiting values.  Since this is so, what's wrong with using other methods to estimate the limiting value, like the classical method?  It is glaringly hypocritical for frequentists like Richard von Mises to condemn all non-frequentist methods as "unscientific," when even his own favored method is itself only capable of estimating the limiting value!   

Misconception #5: The Only Insurable Risks are those for which Relative Frequencies can be Calculated  

One particular misconception about probability deserves special notice here because it has been propagated by one of the foremost minds in the Austrian tradition, Hans-Hermann Hoppe.  Professor Hoppe has been defending the von Mises brothers' conception of probability for some time now, and he has focused particularly on the relationship between probability and insurance.  In his lecture at Mises University in 2011, for example, he adopts a thoroughly Misesian view of probability as it relates to insurance (with a healthy dollop of Frank Knight thrown in for good measure).

Following the lead of Richard von Mises, (who, as was just noted, simply defined the classical method out of existence), Professor Hoppe attempts to rewrite the history of insurance from the frequentist perspective.  Citing Frank Knight and Richard von Mises, he makes the following claim: "Only where the probability calculus can be applied, where we have a collective, where place selection is impossible, only those things are insurable." 

Now, this might not seem to be an objectionable claim, since on one level he is merely making the trivial observation that insurance typically involves a group of people and that it is assumed by all members of the group that any one of them could fall victim to some sort of unforeseeable accident.  Notice, however, that Professor Hoppe claims that only those situations where "the probability calculus" can be applied are insurable.  To say that this is a strange claim is to understate the matter dramatically, because "the probability calculus," (by which he means the calculation of long-run relative frequencies) has only been around for about a century, while the institution of insurance is almost as ancient as man himself!  Indeed, some date the origin of insurance to several thousand years ago, and there have been innumerable insurance institutions between that time and today.  Lloyd's of London alone has been insuring people for hundreds of years. How is it possible for insurance to predate "the probability calculus" of Richard von Mises by thousands of years if Professor Hoppe is right to claim that "What is not controllable through individual actions is insurable, provided long-run objective frequency distributions exist."

The answer is that Professor Hoppe is rewriting the history of insurance from a frequentist perspective.  It would be one thing to simply claim that insurance must involve "classes" of people and a "redistribution of wealth," (although even this claim is dubious, since even Robinson Crusoe can set aside a stockpile of grain on a hilltop to insure himself against hurricanes), but to drag the "probability calculus" of Richard von Mises into the picture is completely bizarre. 

Consider the following hypothetical situation, for example.  You and I decide for whatever reason that our houses are at risk of burning down (because, say, our local fire department is bankrupt and our home insurance companies are hopelessly corrupt), and we decide to set up an insurance arrangement.  We assume (along classical lines) that each of us has an equal likelihood of having our house burn down, and we decide to each set aside a hundred dollars a month in an escrow account to be paid out to the first person whose house burns down.  Our arrangement has absolutely nothing to do with relative frequencies of occurrence or Richard von Mises's "probability calculus."  Nor would our arrangement have anything to do with the calculation of "objective probabilities."  In fact, our arrangement has nothing to do with numerical probability at all!  Does that mean our arrangement is therefore not insurance or that our houses are in principle "uninsurable?"  How could insurance ever have historically arisen, then, since no one prior to the 20th century had any inkling of Richard von Mises's theory of probability? 

The underlying problem here is that Professor Hoppe is following Richard von Mises in treating the relative frequency method as the very definition of probability itself.  The fact that modern-day insurance companies make ample use of relative frequencies in their determination of risk is treated as if this is the critical element in insurance.  The fact that insurance predates Richard von Mises by thousands of years, however, ought to be a pretty obvious sign that Professor Hoppe is misunderstanding the essence of what is going on here.

The essence of insurance is a speculative and subjective bet that assuming some cost today will at least partially offset some possible future contingency.  In the case of modern insurance companies, the insurer makes a subjective assessment of the likelihood of some accident occurring and voluntarily assumes the risks associated with it for a price, while the purchaser of the insurance policy makes a subjective assessment that the price of protection being offered by the insurer is worth the cost.  That's it.  Numerical probability need not have anything to do with this contract.  It is true that insurance companies look at relative frequencies to help them predict the future, but this by no means implies that the risks involved are "objective" in any sense.  Insurance companies are in the business of predicting whether the future will be enough like the past to rely upon past frequencies for guidance.  Such a determination is purely speculative and subjective, and two insurance companies can model future risks in radically different ways even if they have the same relative frequency table from the past sitting in front of them.

If insurance companies have a complete record of the relative frequencies of house fires in New York, for example, this by no means implies that they know what the future relative frequency will be.  They each have to make a prediction about the future relative frequency of occurrence of fires, and this involves subjective judgments about whether the future will be like the past.  The same is true of the people being insured.  If you and I know the complete record of the relative frequencies of house fires in New York, we can nevertheless disagree vehemently about whether the costs of insurance are worth the potential benefit, and whether our personal risks are different from other people's risks.  Past relative frequencies of occurrence need not have anything to do with our judgment about this.  In other words, there is simply no such thing as "objective" risk in the insurance business.

In essence, then, Professor Hoppe is falling victim to the very same error that Merton and Scholes fell victim to in thinking that there exist "objective" risks "out there" in the world to be measured and recorded.  The fact that man can only deal with finite sequences of events, and can never obtain the "limiting values" of those frequencies is simply brushed under the rug.  The possibility that the future can be radically different from the past, and that the relative frequencies we generate can therefore be useless and far from "objective," is completely overlooked.  The caution enjoined on us by men like Nassim Taleb to not confuse past data with "objective" risk is thus, ironically, cast aside by one of the foremost critics of the Chicago School of economics.  Professor Hoppe would have us condemn the methodology of the Chicago School in general, but would join hands with them in claiming that probability is "objective."  Quite strange.

Conclusion

It is my sincere hope that Austrians will start to take the philosophy of probability more seriously than they hitherto have.  It is not enough to simply read Human Action or Probability, Statistics and Truth and think that one has grasped the essence of the issues involved. 

The importance of this issue can scarcely be understated.  Probabilistic modeling has thoroughly infiltrated all of the sciences of human action.  It is important that we Austrians have a more thoughtful response to this infection of probabilistic modeling than simply saying "Well, the relative frequency method can only be applied to 'classes' of events, so all probabilistic modeling of singular human action is nonsense."  Anyone with even a modicum of probabilistic training would respond to this by saying "So what?  All you are saying is that the relative frequency method can only be applied to classes of things.  Duh.  What does this have to do with other probabilistic methods?"

The response that I think Austrians ought to adopt, given their views on causality, is to say that probability is nothing more than a measure of subjective belief.  Coins or dice sitting on a table do not have a mystical property of "objective" probability "in" them.  They are just coins and dice.  The fact that we do not know which side will face upward when they are tossed is due to our own mental limitations and ignorance, not to the coins or dice themselves.  If we knew all of the forces involved in the tossing of a coin or die we would know the outcome beforehand.  The same is true of everything that occurs in the world.  There would be no need for the clumsy and inaccurate methods of probability at all if we knew in advance all of the relevant causal factors involved in any given situation.  Since our ignorance and mental limitations bar us from knowing this, probability is simply a measure of our own subjective uncertainty.     


autor

Mark R. Crovelli
é pos-graduado em ciência política pela Universidade do Colorado.

  • HHH  14/05/2012 16:39
    Dear Mark,
    Sorry, but I consider your views on probability confused and entirely mistaken. Just a few remarks.

    -First I thought it was strange to refer to a lecture of mine instead of my article on the Limits of Numerical Probability in the QJAE.

    On substantive matters:

    - You claim that R. v. Mises defines probability as relative frequency and in so doing rules out any other sense of probability by definition. This is not true. Mises is trying to give a correct theory of probability. He isn't saying that the classical theory is ruled out by definition. Rather, the assumptions that the classical theory makes are wrong.

    - It is also wrong to think that you need to have the correct theoretical account of probability in order to perform probability calculations. This invalidates your point that insurance can't depend on the relative frequency theory, because insurance was around long before that theory was formulated. Of course it was; but so what?

    - You are also wrong that L. v. Mises ignores the possibility of subjective probability estimates. That's precisely the point of Mises's distinction between class and case probability.

    - Cases where individual events are 'insured' are not insurance but simple entrepreneurial ventures based on understanding (case probability) and subject to success/failure.

    - From the fact that objective probabilities can turn out wrong or need to be revised (that humans can err), it does not follow that there are no objective p. and that they are all 'subjective.'

    - Actions (in contrast to natural events) MUST be conceived as unique events (see my mentioned article on this ontological and methodological dualism), and for unique events it is mathematically impossible to formulate numerical probabilities (ratios).


    Best,
    HHH

  • Mark Crovelli  07/04/2013 14:01
    Dear Professor Hoppe,

    I'm very sorry that I never responded to your comment here, but I was only recently advised of its existence. I certainly appreciate your views, and I am glad to have the chance to respond to them.

    I'm afraid that your observations here are all beg the question in precisely the manner that I mention in my articles. You say, for instance, that Richard von Mises was merely concerned "to give a correct theory of probability. It should be obvious, however, that in order for Richard von Mises to do such a thing (or anyone else for that matter), we need to know what probability is in the first place! Pointing out that the assumptions of other methods have limitations does not tell us in any way what probability is. There are also limitations to Richard von Mises's own method, after all, (like the fact that we can never run enough trials to say that we have reached the asymptotic limit), but this by no means tells us what probability is. We need a definition for probability first in order to know whether any of the methods for generating numerical probabilities are justifiable. Since Richard von Mises never offered a definition for probability that did not reference his own frequentist method, there is a very serious circularity problem involved in his project that you do not mention. What you would need to do in order to avoid this circularity problem is to provide us with a definition for probability as an objective property that does not reference the specific mathematical methods of Richard von Mises. Failing this, both you and Richard von Mises are merely begging the question.

    With regard to your second observation, I'm afraid that I don't fully understand. Are you seriously insinuating that insurance companies were indeed utilizing the relative frequency method prior to its invention? If so, I almost don't know how to reply, because it is obvious that insurance companies were simply not using this specific method until very recently.

    As for your comments on Ludwig von Mises's view of probability, I think you are missing the point of my criticism. My criticism is and always has been that the brothers von Mises have been wrong to say that numerical probabilities cannot be applied to singular "cases." To say that Ludwig von Mises allows for subjective probabilities (case probability), and then turn around and say that numerical probabilities cannot be applied to them is to miss the whole point of the debate. It also begs the question, because what we are concerned to know is whether Ludwig was right to make such a claim!

    In summary, I think the burden of proof is on you to provide us with a non-circular definition for probability. I have offered a definition for probability that, I think, is demonstrably consistent with the rest of the Austrian methodological and epistemological system. To the extent that you have provided us with a definition for probability (in the paper you cite), it is wildly question-begging because it merely rehashes the methods of Richard von Mises. Since the legitimacy of his methods are precisely what is at issue, it is question-begging for you to cite them.

    I sincerely hope that you or some other Austrian will attempt to provide us with a non-circular definition for probability, if you really think that my own definition is mistaken.

    Cheers,
    Mark



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