A SINGLE EVENT DOESN’T HAVE A PROBABILITY

A believer in single-event probabilities takes the view that a single flip of a coin or roll of a dice has a probability. I do not. A probability represents the frequency with which an outcome occurs over the very long run, and it is only an average that conceals random variations.

The outcome of a single coin flip can’t be reduced to a percentage or probability. It can only be described in terms of its discrete, mutually exclusive possibilities: heads (H) or tails (T). The outcome of a single roll of a die or pair of dice can only be described in terms of the number of points that may come up, 1 through 6 or 2 through 12.

Yes, the expected frequencies of H, T, and and various point totals can be computed by simple mathematical operations. But those are only expected frequencies. They say nothing about the next coin flip or dice roll, nor do they more than approximate the actual frequencies that will occur over the next 100, 1,000, or 10,000 such events.

Of what value is it to know that the probability of H is 0.5 when H fails to occur in 11 consecutive flips of a fair coin? Of what value is it to know that the probability of rolling a  7 is 0.167 — meaning that 7 comes up every 6 rolls, on average — when 7 may not appear for 56 consecutive rolls? These examples are drawn from simulations of 10,000 coin flips and 1,000 dice rolls. They are simulations that I ran once, not simulations that I cherry-picked from many runs. (The Excel file is at https://drive.google.com/open?id=1FABVTiB_qOe-WqMQkiGFj2f70gSu6a82. Coin flips are at the first tab, dice rolls are at the second tab.)

Let’s take another example, one that is more interesting and has generated much controversy of the years. It’s the Monty Hall problem,

a brain teaser, in the form of a probability puzzle, loosely based on the American television game show Let’s Make a Deal and named after its original host, Monty Hall. The problem was originally posed (and solved) in a letter by Steve Selvin to the American Statistician in 1975…. It became famous as a question from a reader’s letter quoted in Marilyn vos Savant’s “Ask Marilyn” column in Parade magazine in 1990 … :

Suppose you’re on a game show, and you’re given the choice of three doors: Behind one door is a car; behind the others, goats. You pick a door, say No. 1, and the host, who knows what’s behind the doors, opens another door, say No. 3, which has a goat. He then says to you, “Do you want to pick door No. 2?” Is it to your advantage to switch your choice

Vos Savant’s response was that the contestant should switch to the other door…. Under the standard assumptions, contestants who switch have a 2/3 chance of winning the car, while contestants who stick to their initial choice have only a 1/3 chance.

Vos Savant’s answer is correct, but only if the contestant is allowed to play an unlimited number of games. A player who adopts a strategy of “switch” in every game will, in the long run, win about 2/3 of the time (explanation here). That is, the player has a better chance of winning if he chooses “switch” rather than “stay”.

Read the preceding paragraph carefully and you will spot the logical defect that underlies the belief in single-event probabilities: The long-run winning strategy (“switch”) is transformed into a “better chance” to win a particular game. What does that mean? How does an average frequency of 2/3 improve one’s chances of winning a particular game? It doesn’t. Game results are utterly random; that is, the average frequency of 2/3 has no bearing on the outcome of a single game.

I’ll try to drive the point home by returning to the coin-flip game, with money thrown into the mix. A $1 bet on H means a gain of $1 if H turns up, and a loss of $1 if T turns up. The expected value of the bet — if repeated over a very large number of trials — is zero. The bettor expects to win and lose the same number of times, and to walk away no richer or poorer than when he started. And for a very large number of games, the better will walk away approximately (but not necessarily exactly) neither richer nor poorer than when he started. How many games? In the simulation of 10,000 games mentioned earlier, H occurred 50.6 percent of the time. A very large number of games is probably at least 100,000.

Let us say, for the sake of argument, that a bettor has played 100,00 coin-flip games at $1 a game and come out exactly even. What does that mean for the play of the next game? Does it have an expected value of zero?

To see why the answer is “no”, let’s make it interesting and say that the bet on the next game — the next coin flip — is $10,000. The size of the bet should wonderfully concentrate the bettor’s mind. He should now see the situation for what it really is: There are two possible outcomes, and only one of them will be realized. An average of the two outcomes is meaningless. The single coin flip doesn’t have a “probability” of 0.5 H and 0.5 T and an “expected payoff” of zero. The coin will come up either H or T, and the bettor will either lose $10,000 or win $10,000.

To repeat: The outcome of a single coin flip doesn’t have an expected value for the bettor. It has two possible values, and the bettor must decide whether he is willing to lose $10,000 on the single flip of a coin.

By the same token (or coin), the outcome of a single roll of a pair of dice doesn’t have a 1-in-6 probability of coming up 7. It has 36 possible outcomes and 11 possible point totals, and the bettor must decide how much he is willing to lose if he puts his money on the wrong combination or outcome.

In summary, it is a logical fallacy to ascribe a probability to a single event. A probability represents the observed or computed average value of a very large number of like events. A single event cannot possess that average value. A single event has a finite number of discrete and mutually exclusive possible outcomes. Those outcomes will not “average out” in that single event. Only one of them will obtain, like Schrödinger’s cat.

To say or suggest that the outcomes will average out — which is what a probability implies — is tantamount to saying that Jack Sprat and his wife were neither skinny nor fat because their body-mass indices averaged to a normal value. It is tantamount to saying that one can’t drown by walking across a pond with an average depth of 1 foot, when that average conceals the existence of a 100-foot-deep hole.

It should go without saying that a specific event that might occur — rain tomorrow, for example — doesn’t have a probability.

WHAT ABOUT THE PROBABILITY OF PRECIPITATION?

Weather forecasters (meteorologists) are constantly saying things like “there’s an 80-percent probability of precipitation (PoP) in __________ tomorrow”. What do such statements mean? Not much:

It is not surprising that this issue is difficult for the general public, given that it is debated even within the scientific community. Some propose a “frequentist” interpretation: there will be at least a minimum amount of rain on 80% of days with weather conditions like they are today. Although preferred by many scientists, this explanation may be particularly difficult for the general public to grasp because it requires regarding tomorrow as a class of events, a group of potential tomorrows. From the perspective of the forecast user, however, tomorrow will happen only once. A perhaps less abstract interpretation is that PoP reflects the degree of confidence that the forecaster has that it will rain. In other words, an 80% chance of rain means that the forecaster strongly believes that there will be at least a minimum amount of rain tomorrow. The problem, from the perspective of the general public, is that when PoP is forecasted, none of these interpretations is specified.

There are clearly some interpretations that are not correct. The percentage expressed in PoP neither refers directly to the percent of area over which precipitation will fall nor does it refer directly to the percent of time precipitation will be observed on the forecast day Although both interpretations are clearly wrong, there is evidence that the general public holds them to varying degrees. Such misunderstandings are critical because they may affect the decisions that people make. If people misinterpret the forecast as percent time or percent area, they maybe more inclined to take precautionary action than are those who have the correct probabilistic interpretation, because they think that it will rain somewhere or some time tomorrow. The negative impact of such misunderstandings on decision making, both in terms of unnecessary precautions as well as erosion in user trust, could well eliminate any potential benefit of adding uncertainty information to the forecast. [Susan Joslyn, Nimor Nadav-Greenberg, and Rebecca M. Nichols, “Probability of Precipitations: Assessment and Enhancement of End-User Understanding“, Journal of the American Meteorological Society, February 2009, citations omitted]

The frequentist interpretation is close to be correct, but it still involves a great deal of guesswork. Rainfall in a particular location is influenced by many variables (e.g., atmospheric pressure, direction and rate of change of atmospheric pressure, ambient temperature, local terrain, presence or absence of bodies of water, vegetation, moisture content of the atmosphere, height of clouds above the terrain, depth of cloud cover). It is nigh unto impossible to say that today’s (or tomorrow’s or next week’s) weather conditions are like (or will be like) those that in the past resulted in rainfall in a particular location 80 percent of the time.

That leaves the Bayesian interpretation, in which the forecaster combines some facts (e.g., the presence or absence of a low-pressure system in or toward the area, the presence or absence of a flow of water vapor in or toward the area) with what he has observed in the past to arrive at a guess about future weather. He then attaches a probability to his guess to indicate the strength of his confidence in it.

Thus:

Bayesian probability represents a level of certainty relating to a potential outcome or idea. This is in contrast to a frequentist probability that represents the frequency with which a particular outcome will occur over any number of trials.

An event with Bayesian probability of .6 (or 60%) should be interpreted as stating “With confidence 60%, this event contains the true outcome”, whereas a frequentist interpretation would view it as stating “Over 100 trials, we should observe event X approximately 60 times.”

And thus:

The Bayesian approach to learning is based on the subjective interpretation of probability. The value of the proportion p is unknown, and a person expresses his or her opinion about the uncertainty in the proportion by means of a probability distribution placed on a set of possible values of p.

It is impossible to attach a probability — as properly defined in the first part of this article — to something that hasn’t happened, and may not happen. So when you read or hear a statement like “the probability of rain tomorrow is 80 percent”, you should mentally translate it into language like this:

X guesses that Y will (or will not) happen at time Z, and the “probability” that he attaches to his guess  indicates his degree of confidence in it.

The guess may be well-informed by systematic observation of relevant events, but it remains a guess. As most Americans have learned and relearned over the years, when rain has failed to materialize or has spoiled an outdoor event that was supposed to be rain-free.

BUT AREN’T SOME THINGS MORE LIKELY TO HAPPEN THAN OTHERS?

Of course. But only one thing will happen at a given time and place.

If a person walks across a shooting range where live ammunition is being used, his is more likely to be killed than if he walks across the same patch of ground when no one is shooting. And a clever analyst could concoct a probability of a person’s being shot by writing an equation that includes such variables as his size, the speed with which he walks, the number of shooters, their rate of fire, and the distance across the shooting range.

What would the probability estimate mean? It would mean that if a very large number of persons walked across the shooting range under identical conditions, approximately S percent of them would be shot. But the clever analyst cannot specify which of the walkers would be among the S percent.

Here’s another way to look at it. One person wearing head-to-toe bullet-proof armor could walk across the range a large number of times and expect to be hit by a bullet on S percent of his crossings. But the hardy soul wouldn’t know on which of the crossings he would be hit.

Suppose the hardy soul became a foolhardy one and made a bet that he could cross the range without being hit. Further, suppose that S is estimated to be 0.75; that is, 75 percent of a string of walkers would be hit, or a single (bullet-proof) walker would be hit on 75 percent of his crossings. Knowing the value of S, the foolhardy fellow offers to pay out $1 million dollars if he crosses the range unscathed — one time — and claim $4 million (for himself or his estate) if he is shot. That’s an even-money bet, isn’t it?

No it isn’t. This situation is exactly analogous to the $10,000 bet on a single coin flip, discussed above. But I will dissect this one in a different way, to the same end.

The bet should be understood for what it is, an either-or-proposition. The foolhardy walker will either lose $1 million or win $4 million. The bettor (or bettors) who take the other side of the bet will either win $1 million or lose $4 million.

As anyone with elementary reading and reasoning skills should be able to tell, those possible outcomes are not the same as the outcome that would obtain (approximately) if the foolhardy fellow could walk across the shooting range 1,000 times. If he could, he would come very close to breaking even, as would those who bet against him.

To put it as simply as possible:

When an event has more than one possible outcome, a single trial cannot replicate the average outcome of a large number of trials (replications of the event).

It follows that the average outcome of a large number of trials — the probability of each possible outcome — cannot occur in a single trial.

It is therefore meaningless to ascribe a probability to any possible outcome of a single trial.

MODELING AND PROBABILITY

Sometimes, when things interact, the outcome of the interactions will conform to an expected value — if that value is empirically valid. For example, if a pot of pure water is put over a flame at sea level, the temperature of the water will rise to 212 degrees Fahrenheit and water molecules will then begin to escape into the air in a gaseous form (steam).  If the flame is kept hot enough and applied long enough, the water in the pot will continue to vaporize until the pot is empty.

That isn’t a probabilistic description of boiling. It’s just a description of what’s known to happen to water under certain conditions.

But it bears a similarity to a certain kind of probabilistic reasoning. For example, in a paper that I wrote long ago about warfare models, I said this:

Consider a five-parameter model, involving the conditional probabilities of detecting, shooting at, hitting, and killing an opponent — and surviving, in the first place, to do any of these things. Such a model might easily yield a cumulative error of a hemibel [a factor of 3], given a twenty five percent error in each parameter.

Mathematically, 1.25⁵ = 3.05. Which is true enough, but also misleadingly simple.

A mathematical model of that kind rests on the crucial assumption that the component probabilities are based on observations of actual events occurring in similar conditions. It is safe to say that the values assigned to the parameters of warfare models, econometric models, sociological models, and most other models outside the realm of physics, chemistry, and other “hard” sciences fail to satisfy that assumption.

Further, a mathematical model yields only the expected (average) outcome of a large number of events occurring under conditions similar to those from which the component probabilities were derived. (A Monte Carlo model merely yields a quantitative estimate of the spread around the average outcome.) Again, this precludes most models outside the “hard” sciences, and even some within that domain.

The moral of the story: Don’t be gulled by a statement about the expected outcome of an event, even when the statement seems to be based on a rigorous mathematical formula. Look behind the formula for an empirical foundation. And not just any empirical foundation, but one that is consistent with the situation to which the formula is being applied.

And when you’ve done that, remember that the formula expresses a point estimate around which there’s a wide — very wide — range of uncertainty. Which was the real point of the passage quoted above. The only sure things in life are death, taxes, and regulation.

PROBABILITY VS. OPPORTUNITY

Warfare models, as noted, deal with interactions among large numbers of things. If a large unit of infantry encounters another large unit of enemy infantry, and the units exchange gunfire, it is reasonable to expect the following consequences:

• As the numbers of infantrymen increase, more of them will be shot, for a given rate of gunfire.

• As the rate of gunfire increases, more of the infantrymen will be shot, for a given number of infantrymen.

These consequences don’t represent probabilities, though an inveterate modeler will try to represent them with a probabilistic model. They represent opportunities — opportunities for bullets to hit bodies. It is entirely possible that some bullets won’t hit bodies and some bodies won’t be hit by bullets. But more bullets will hit bodies if there are more bodies in a given space. And a higher proportion of a given number of bodies will be hit as more bullets enter a given space.

That’s all there is to it.

It has nothing to do with probability. The actual outcome of a past encounter is the actual outcome of that encounter, and the number of casualties has everything to do with the minutiae of the encounter and nothing to do with probability. A fortiori, the number of casualties resulting from a possible future encounter would have everything to do with the minutiae of that encounter and nothing to do with probability. Given the uniqueness of any given encounter, it would be wrong to characterize its outcome (e.g., number of casualties per infantryman) as a probability.

Related posts:
Understanding the Monty Hall Problem
The Compleat Monty Hall Problem
Some Thoughts about Probability
My War on the Misuse of Probability
Scott Adams Understands Probability
Further Thoughts about Probability

In the preceding post I say that “the problem with history is that the future isn’t part of it.” That is subtle criticism of the too-frequent practice of attributing a probability to the occurrence of a future event — especially a unique event, such as a war here or a terrorist attack there.

A probability is a statement about a very large number of like events, each of which has an unpredictable (random) outcome. Probability, properly understood, says nothing about the outcome of an individual event. It certainly says nothing about what will happen next.

A fair coin comes up heads with a probability of 0.5, and comes up tails with the same probability. But those aren’t statements about the outcome of the next coin toss. No, they’re statements about the approximate frequencies of the occurrence of heads and tails in a large number of tosses. The next coin toss will eventuate in heads or tails, but not 0.5 heads and 0.5 tails (except in the rare and unpredictable case of a coin landing on edge and staying there).

There’s a vast gap between routine processes of the kind to which probabilities attach — coin tosses, for example — and the complexities of human activity. Human activity is too complex and dependent on intentions and willful actions to be characterized (properly) by statements about the probability of this or that action.

It is fatuous to say, for example, that a war on the scale of World War II is improbable because such a war has occurred only once in human history. By that reasoning, one could have said confidently in 1938 that a war on the scale of World War II could never occur because there had been no such war in human history.

(Inspired by Bryan Caplan’s fatuous post, “So Far.”)

Here.

REVISED 02/09/15 WITH AN ADDENDUM AT THE END OF THE POST

This post is prompted by a reader’s comments about “The Compleat Monty Hall Problem.” I open with a discussion of probability and its inapplicability to single games of chance (e.g., one toss of a coin). With that as background, I then address the reader’s specific comments. I close with a discussion of the debasement of the meaning of probability.

INTRODUCTORY REMARKS

What is probability? Is it a property of a thing (e.g., a coin), a property of an event involving a thing (e.g., a toss of the coin), or a description of the average outcome of a large number of such events (e.g., “heads” and “tails” will come up about the same number of times)? I take the third view.

What does it mean to say, for example, that there’s a probability of 0.5 (50 percent) that a tossed coin will come up “heads” (H), and a probability of 0.5 that it will come up “tails” (T)? Does such a statement have any bearing on the outcome of a single toss of a coin? No, it doesn’t. The statement is only a short way of saying that in a sufficiently large number of tosses, approximately half will come up H and half will come up T. The result of each toss, however, is a random event — it has no probability.

That is the standard, frequentist interpretation of probability, to which I subscribe. It replaced the classical interpretation , which is problematic:

If a random experiment can result in N mutually exclusive and equally likely outcomes and if N_A of these outcomes result in the occurrence of the event A, the probability of A is defined by

.

There are two clear limitations to the classical definition.^[16] Firstly, it is applicable only to situations in which there is only a ‘finite’ number of possible outcomes. But some important random experiments, such as tossing a coin until it rises heads, give rise to an infinite set of outcomes. And secondly, you need to determine in advance that all the possible outcomes are equally likely without relying on the notion of probability to avoid circularity….

A similar charge has been laid against frequentism:

It is of course impossible to actually perform an infinity of repetitions of a random experiment to determine the probability of an event. But if only a finite number of repetitions of the process are performed, different relative frequencies will appear in different series of trials. If these relative frequencies are to define the probability, the probability will be slightly different every time it is measured. But the real probability should be the same every time. If we acknowledge the fact that we only can measure a probability with some error of measurement attached, we still get into problems as the error of measurement can only be expressed as a probability, the very concept we are trying to define. This renders even the frequency definition circular.

Not so:

• There is no “real probability.” If there were, the classical theory would measure it, but the classical theory is circular, as explained above.

• It is therefore meaningless to refer to “error of measurement.” Estimates of probability may well vary from one series of trials to another. But they will “tend to a fixed limit” over many trials (see below).

There are other approaches to probability. See, for example, this, this, and this.) One approach is known as propensity probability:

Propensities are not relative frequencies, but purported causes of the observed stable relative frequencies. Propensities are invoked to explain why repeating a certain kind of experiment will generate a given outcome type at a persistent rate. A central aspect of this explanation is the law of large numbers. This law, which is a consequence of the axioms of probability, says that if (for example) a coin is tossed repeatedly many times, in such a way that its probability of landing heads is the same on each toss, and the outcomes are probabilistically independent, then the relative frequency of heads will (with high probability) be close to the probability of heads on each single toss. This law suggests that stable long-run frequencies are a manifestation of invariant single-case probabilities.

This is circular. You observe the relative frequencies of outcomes and, lo and behold, you have found the “propensity” that yields those relative frequencies.

Another approach is Bayesian probability:

Bayesian probability represents a level of certainty relating to a potential outcome or idea. This is in contrast to a frequentist probability that represents the frequency with which a particular outcome will occur over any number of trials.

An event with Bayesian probability of .6 (or 60%) should be interpreted as stating “With confidence 60%, this event contains the true outcome”, whereas a frequentist interpretation would view it as stating “Over 100 trials, we should observe event X approximately 60 times.”

Or consider this account:

The Bayesian approach to learning is based on the subjective interpretation of probability.   The value of the proportion p is unknown, and a person expresses his or her opinion about the uncertainty in the proportion by means of a probability distribution placed on a set of possible values of p….

“Level of certainty” and “subjective interpretation” mean “guess.” The guess may be “educated.” It’s well known, for example, that a balanced coin will come up heads about half the time, in the long run. But to say that “I’m 50-percent confident that the coin will come up heads” is to say nothing meaningful about the outcome of a single coin toss. There are as many probable outcomes of a coin toss as there are bystanders who are willing to make a statement like “I’m x-percent confident that the coin will come up heads.” Which means that a single toss doesn’t have a probability, though it can be the subject of many opinions as to the outcome.

Returning to reality, Richard von Mises eloquently explains frequentism in Probability, Statistics and Truth (second revised English edition, 1957). Here are some excerpts:

The rational concept of probability, which is the only basis of probability calculus, applies only to problems in which either the same event repeats itself again and again, or a great number of uniform elements are involved at the same time. Using the language of physics, we may say that in order to apply the theory of probability we must have a practically unlimited sequence of uniform observations. [P. 11]

*     *     *

In games of dice, the individual event is a single throw of the dice from the box and the attribute is the observation of the number of points shown by the dice. In the game of “heads or tails”, each toss of the coin is an individual event, and the side of the coin which is uppermost is the attribute. [P. 11]

*     *     *

We must now introduce a new term…. This term is “the collective”, and it denotes a sequence of uniform events or processes which differ by certain observable attributes…. All the throws of dice made in the course of a game [of many throws] from a collective wherein the attribute of the single event is the number of points thrown…. The definition of probability which we shall give is concerned with ‘the probability of encountering a single attribute in a given collective’. [Pp. 11-12]

*     *     *

[A] collective is a mass phenomenon or a repetitive event, or, simply, a long sequence of observations for which there are sufficient reasons to believe that the relative frequency of the observed attribute would tend to a fixed limit if the observations were indefinitely continued. The limit will be called the probability of the attribute considered within the collective. [P. 15, emphasis in the original]

*     *     *

The result of each calculation … is always … nothing else but a probability, or, using our general definition, the relative frequency of a certain event in a sufficiently long (theoretically, infinitely long) sequence of observations. The theory of probability can never lead to a definite statement concerning a single event. The only question that it can answer is: what is to be expected in the course of a very long sequence of observations? [P. 33, emphasis added]

As stated earlier, it is simply meaningless to say that the probability of H or T coming up in a single toss is 0.5. Here’s the proper way of putting it: There is no reason to expect a single coin toss to have a particular outcome (H or T), given that the coin is balanced, the toss isn’t made is such a way as to favor H or T, and there are no other factors that might push the outcome toward H or T. But to say that P(H) is 0.5 for a single toss is to misrepresent the meaning of probability, and to assert something meaningless about a single toss.

If you believe that probabilities attach to a single event, you must also believe that a single event has an expected value. Let’s say, for example, that you’re invited to toss a coin once, for money. You get $1 if H comes up; you pay $1 if T comes up. As a believer in single-event probabilities, you “know” that you have a “50-50 chance” of winning or losing. Would you play a single game, which has an expected value of $0? If you would, it wouldn’t be because of the expected value of the game; it would be because you might win $1, and because losing $1 would mean little to you.

Now, change the bet from $1 to $1,000. The “expected value” of the single game remains the same: $0. But the size of the stake wonderfully concentrates your mind. You suddenly see through the “expected value” of the game. You are struck by the unavoidable fact that what really matters is the prospect of winning $1,000 or losing $1,000, because those are the only possible outcomes.

Your decision about playing a single game for $1,000 will depend on your finances (e.g., you may be very wealthy or very desperate for money) and your tolerance for risk (e.g., you may be averse to risk-taking or addicted to it). But — if you are rational — you will not make your decision on the basis of the fictional expected value of a single game, which derives from the fictional single-game probabilities of H and T. You will decide whether you’re willing and able to risk the loss of $1,000.

Do I mean to say that probability is irrelevant to a single play of the Monty Hall problem, or to a choice between games of chance? If you’re a proponent of propensity, you might say that in the Monty Hall game the prize has a propensity to be behind the other unopened door (i.e., the door not chosen by you and not opened by the host). But does that tell you anything about the actual location of the prize in a particular game? No, because the “propensity” merely reflects the outcomes of many games; it says nothing about a single game, which (like Schrödinger’s cat) can have only a single outcome (prize or no prize), not 2/3 of one.

If you’re a proponent of Bayesian probability, you might say that you’re confident with “probability” 2/3 that the prize is behind the other unopened door. But that’s just another way of saying that contestants win 2/3 of the time if they always switch doors. That’s the background knowledge that you bring to your statement of confidence. But someone who’s ignorant of the Monty Hall problem might be confident with 1/2 “probability” that the prize is behind the other unopened door. And he could be right about a particular game, despite his lower level of confidence.

So, yes, I do mean to say that there’s no such thing as a single-case probability. You may have an opinion ( or a hunch or a guess) about the outcome of a single game, but it’s only your opinion (hunch, guess). In the end, you have to bet on a discrete outcome. If it gives you comfort to switch to the unopened door because that’s the winning door 2/3 of the time (according to classical probability) and about 2/3 of the time (according to the frequentist interpretation), be my guest. I might do the same thing, for the same reason: to be comfortable about my guess. But I’d be able separate my psychological need for comfort from the reality of the situation:

A single game is just one event in the long series of events from which probabilities emerge. I can win the Monty Hall game about 2/3 of the time in repeated plays if I always switch doors. But that probability has nothing to do with a single game, the outcome of which is a random occurrence.

REPLIES TO A READER’S COMMENTS

I now turn to the reader’s specific comments, which refer to “The Compleat Monty Hall Problem.” (You should read it before continuing with this post if you’re unfamiliar with the Monty Hall problem or my analysis of it.) The reader’s comments — which I’ve rearranged slightly — are in italic type. (Here and there, I’ve elaborated on the reader’s comments; my elaborations are placed in brackets and set in roman type.) My replies are in bold type.

I find puzzling your statement that a probability cannot “describe” a single instance, eg one round of the Monty Hall problem.

See my introductory remarks.

While the long run result serves to prove the probability of a particular outcome, that does not mean that that probability may not be assigned to a smaller number of instances. That is the beauty of probability.

The long-run result doesn’t “prove” the probability of a particular outcome; it determines the relative frequency of occurrence of that outcome — and nothing more. There is no probability associated with a “smaller number of instances,” certainly not 1 instance. Again, see my introductory remarks.

If the [Monty Hall] game is played once [and I don’t switch doors], I should budget for one car [the prize that’s usually cited in discussions of the Monty hall problem], and if it is played 100 times [and I never switch doors], I budget for 33….

“Budget” seems to refer to the expected number of cars won, given the number of plays of the game and a strategy of never switching doors. The reader contradicts himself by “budgeting” for 1 car in a single play of the Monty Hall problem. In doing so, he is being unfaithful to his earlier statement: “While the long run result serves to prove the probability of a particular outcome, that does not mean that that probability may not be assigned to a smaller number of instances.” Removing the double negatives, we get “probability may be assigned to a smaller number of instances.” Given that 1 is a smaller number than 100, it follows, by the reader’s logic, that his “budget” for a single game should be 1/3 car (assuming, as he does, a strategy of not switching doors). The reader’s problem here is his insistence that a probability expresses something other than the long-run relative frequency of a particular outcome.

To justify your contrary view, you ask how you can win 2/3 of a car [the long-run average if the contestant plays many games and always switches doors]; you can win or you can not win, you say, you cannot partly win. Is this not sophistry or a straw man, sloppy reasoning at best, to convince uncritical thinkers who agree that you cannot drive 2/3 of a car?

My “contrary view” of what? My view of statistics isn’t “contrary.” Rather, it’s in line with the standard, frequentist interpretation.

It’s a simple statement of obvious fact you can’t win 2/3 of a car. There’s no “sophistry” or “straw man” about it. If you can’t win 2/3 of a car, what does it mean to assign a probability of 2/3 to winning a car by adopting the switching strategy? As discussed above, it means only one thing: A long series of games will be won about 2/3 of the time if all contestants adopt the switching strategy.

On what basis other than an understanding of probability would you be optimistic at the prospect of being offered one chance of picking a single Golden Ball worth $1m from a bag of just three balls and pessimistic about your prospects of picking the sole Golden Ball from a barrel of 10,000 balls?

The only difference between the two games is that on the one hand you have a decent (33%) chance of winning and on the other hand you have a lousy (0.01%) chance. Isn’t it these disparate probabilities that give you cause for optimism or pessimism, as the case may be?

“Optimism” and “pessimism” — like “comfort” — are subjective terms for ill-defined states of mind. There are persons who will be “optimistic” about a given situation, and persons who will be “pessimistic” about the same situation. For example: There are hundreds of millions of persons who are “optimistic” about winning various lotteries, even though they know that the grand prize in each lottery will be assigned to only one of millions of possible numbers. By the same token, there are hundreds of millions of persons who, knowing the same facts, refuse to buy lottery tickets because they are “pessimistic” about the likely outcome of doing so. But “optimism” and “pessimism” — like “comfort” — have nothing to do with probability, which isn’t an attribute of a single game.

If probability cannot describe the chances of each of the two one-off “games”, does that mean I could not provide a mathematical basis for my advice that you play the game with 3 balls (because you have a one-in-three chance of winning) rather than the ball in the barrel game which offers a one in ten thousand chance of winning?

You can provide a mathematical basis for preferring the game with 3 balls. But you must, in honesty, state that the mathematical basis applies only to many games, and that the outcome of a single game is unpredictable.

It might be that probability cannot reliably describe the actual outcome of a single event because the sample size of 1 game is too small to reflect the long-run average that proves the probability. However, comparing the probabilities for winning the two games describes the relative likelihood of winning each game and informs us as to which game will more likely provide the prize.

If not by comparing the probability of winning each game, how do we know which of the two games has a better chance of delivering a win? One cannot compare the probability of selecting the Golden Ball from each of the two games unless the probability of each game can be expressed, or described, as you say.

Here, the reader comes close to admitting that a probability can’t describe the (expected) outcome of a single event (“reliably” is superfluous). But he goes off course when he says that “comparing the probabilities for the two games … informs us as to which game will more likely provide the prize.” That statement is true only for many plays of the two ball games. It has nothing to do with a single play of either ball game. The choice there must be based on subjective considerations: “optimism,” “pessimism,” “comfort,” a guess, a hunch, etc.

Can I not tell a smoker that their lifetime risk of developing lung cancer is 23% even though smokers either get lung cancer or they do not? No one gets 23% cancer. Did someone say they did? No one has 0.2 of a child either but, on average, every family in a census did at one stage have 2.2 children.

No, the reader may not (honestly) tell a smoker that his lifetime risk of developing lung cancer is 23 percent, or any specific percentage. The smoker has one life to live; he will either get lung cancer or he will not. What the reader may honestly tell the smoker is that statistics based on the fates of a large number of smokers over many decades indicate that a certain percentage of those smokers contracted lung cancer. The reader should also tell the smoker that the frequency of the incidence of lung cancer in a large population varies according to the number of cigarettes smoked daily. (According to Wikipedia: “For every 3–4 million cigarettes smoked, one lung cancer death occurs.^[1][132]“) Further, the reader should note that the incidence of lung cancer also varies with the duration of smoking at various rates, and with genetic and environmental factors that vary from person to person.

As for family size, given that the census counts only post-natal children (who come in integer values), how could “every family in a census … at one stage have 2.2 children”? The average number of children across a large number of families may be 2.2, but surely the reader knows that “every family” did not somehow have 2.2 children “at one stage.” And surely the reader knows that average family size isn’t a probabilistic value, one that measures the relative frequency of an event (e.g., “heads”) given many repetitions of the same trial (e.g., tossing a fair coin), under the same conditions (e.g., no wind blowing). Each event is a random occurrence within the long string of repetitions. The reader may have noticed that family size is in fact strongly determined (especially in Western countries) by non-random events (e.g., deliberate decisions by couples to reproduce, or not). In sum, probabilities may represent averages, but not all (or very many) averages represent probabilities.

If not [by comparing probabilities], how do we make a rational recommendation and justify it in terms the board of a think-tank would accept? [This seems to be a reference to my erstwhile position as an officer of a defense think-tank.]

Here, the reader extends an inappropriate single-event view of probability to an inappropriate unique-event view. I would not have gone before the board and recommended a course of action — such as bidding on a contract for a new line of work — based on a “probability of success.” That would be an absurd statement to make about an event that is defined by unique circumstances (e.g., the composition of the think-tank’s staff at that time, the particular kind of work to be done, the qualifications of prospective competitors’ staffs). I would simply have spelled out the facts and the uncertainties. And if I had a hunch about the likely success or failure of the venture, I would have recommended for or against it, giving specific reasons for my hunch (e.g., the relative expertise of our staff and competitors’ staffs). But it would have been nothing more than a hunch; it wouldn’t have been my (impossible) assessment of the probability of a unique event.

Boards (and executives) don’t base decisions on (non-existent) probabilities; they base decisions on unique sets of facts, and on hunches (preferably hunches rooted in knowledge and experience). Those hunches may sometimes be stated as probabilities, as in “We’ve got a 50-50 chance of winning the contract.” (Though I would never say such a thing.) But such statements are only idiomatic, and have nothing to do with probability as it is properly understood.

CLOSING THOUGHTS

The reader’s comments reflect the popular debasement of the meaning of probability. The word has been adapted to many inappropriate uses: the probability of precipitation (a quasi-subjective concept), the probability of success in a business venture (a concept that requires the repetition of unrepeatable events), the probability that a batter will get a hit in his next at-bat (ditto, given the many unique conditions that attend every at-bat), and on and on. The effect of all such uses (and, often, the purpose of such uses) is to make a guess seem like a “scientific” prediction.

ADDENDUM (02/09/15)

The reader whose comments about “The Compleat Monty Hall Problem” I address above has submitted some additional comments.

The first additional comment pertains to this exchange where the reader’s remarks are in italics and my reply is in bold:

If the [Monty Hall] game is played once [and I don’t switch doors], I should budget for one car [the prize that’s usually cited in discussions of the Monty hall problem], and if it is played 100 times [and I never switch doors], I budget for 33….

“Budget” seems to refer to the expected number of cars won, given the number of plays of the game and a strategy of never switching doors. The reader contradicts himself by “budgeting” for 1 car in a single play of the Monty Hall problem. In doing so, he is being unfaithful to his earlier statement: “While the long run result serves to prove the probability of a particular outcome, that does not mean that that probability may not be assigned to a smaller number of instances.” Removing the double negatives, we get “probability may be assigned to a smaller number of instances.” Given that 1 is a smaller number than 100, it follows, by the reader’s logic, that his “budget” for a single game should be 1/3 car (assuming, as he does, a strategy of not switching doors). The reader’s problem here is his insistence that a probability expresses something other than the long-run relative frequency of a particular outcome.

The reader’s rejoinder (with light editing by me):

The game-show producer “budgets” one car if playing just one game because losing one car is a possible outcome and a prudent game-show producer would cover all possibilities, the loss of one car being one of them.  This does not contradict anything I have said, it is simply the necessary approach to manage the risk of having a winning contestant and no car.  Similarly, the producer budgets 33 cars for a 100-show season [TEA: For consistency, “show” should be “game”].

The contradiction is between the reader’s use of an expected-value calculation for 100 games, but not for a single game. If the game-show producer knows (how?) that contestants will invariably stay with the doors they’ve chosen initially, a reasonable budget for a 100-game season is 33 cars. (But a “reasonable” budget isn’t a foolproof one, as I show below.) By the same token, a reasonable budget for a single game — a game played only once, not one game in a series — is 1/3 car. After all, that is the probabilistic outcome of a single game if you believe that a probability can be assigned to a single game. And the reader does believe that; here’s the first sentence of his original comments:

I find puzzling your statement that a probability cannot “describe” a single instance, eg one round of the Monty Hall problem. [See the section “Replies to a Reader’s Comments” in “Some Thoughts about Probability.”]

Thus, according the reader’s view of probability, the game-show producer should budget for 1/3 car. After all, in the reader’s view, there’s a 2/3 probability that a contestant won’t win a car in a one-game run of the show.

The reader could respond that cars come in units of one. True, but the designation of a car as the prize is arbitrary (and convenient for the reader). The prize could just as well be money — $30,000 for example. If the contestant wins a car in the (rather contrived) one-game run of the show, the producer then (and only then) gives the contestant a check for $30,000. But, by the reader’s logic, the game has two other equally likely outcomes: the contestant loses and the contestant loses. If those outcomes prevail, the producer doesn’t have to write a check. So, the average prize for the one-game run of the show would be $10,000, or the equivalent of 1/3 car.

Now, the producer might hedge his bets because the outcome of a single game is uncertain; that is, he might budget one car or $30,000. But by the same logic, the producer should budget 100 cars or $3,000,000 for 100 games, not 33 cars or $990,000. Again, the reader contradicts himself. He uses an expected-value calculation for one game but not for 100 games.

What is a “reasonable” budget for 100 games, or fewer than 100 games? Well, it’s really a subjective call that the producer must make, based on his tolerance for risk. The producer who budgets 33 cars or $990,000 for 100 games on the basis of an expected-value calculation may find himself without a job.

Using a random-number generator, I set up a simulation of the outcomes of games 1 through 100, where the contestant always stays with the door originally chosen . Here are the results of the first five simulations that I ran:

Some observations:

Outcomes of individual games are unpredictable, as evidenced by the wild swings and jagged lines, the latter of which persist even at 100 games.

Taking the first game as a proxy for a single-game run of the show, we see that the contestant won that game in just one of the five simulations. To put it another way, in four of the five cases the producer would have thrown away his money on a rapidly depreciating asset (a car) if had offered a car as the prize.

Results vary widely and wildly after the first game. At 10 and 20 games, contestants are doing better than expected in four of the five simulations. At 30 games, contestants are doing as well or better than expected in all five simulations. At 100 games, contestants are doing better than expected in two simulation, worse than expected in two simulation, and exactly as expected in one simulation. What should the producer do with such information? Well, it’s up to the producer and his tolerance for risk. But a prudent producer wouldn’t budget 33 cars or $990,000 just because that’s the expected value of 100 games.

It can take a lot of games to yield an outcome that comes close to 1/3 car per game. A game-show producer could easily lose his shirt by “betting” on 1/3 car per game for a season much shorter than 100 games, and even for a season of 100 games.

The reader’s second additional comment pertains to this exchange:

On what basis other than an understanding of probability would you be optimistic at the prospect of being offered one chance of picking a single Golden Ball worth $1m from a bag of just three balls and pessimistic about your prospects of picking the sole Golden Ball from a barrel of 10,000 balls?

The only difference between the two games is that on the one hand you have a decent (33%) chance of winning and on the other hand you have a lousy (0.01%) chance. Isn’t it these disparate probabilities that give you cause for optimism or pessimism, as the case may be?

“Optimism” and “pessimism” — like “comfort” — are subjective terms for ill-defined states of mind. There are persons who will be “optimistic” about a given situation, and persons who will be “pessimistic” about the same situation. For example: There are hundreds of millions of persons who are “optimistic” about winning various lotteries, even though they know that the grand prize in each lottery will be assigned to only one of millions of possible numbers. By the same token, there are hundreds of millions of persons who, knowing the same facts, refuse to buy lottery tickets because they are “pessimistic” about the likely outcome of doing so. But “optimism” and “pessimism” — like “comfort” — have nothing to do with probability, which isn’t an attribute of a single game.

The reader now says this:

Optimism or pessimism are states as much as something being hot or cold, or a person perspiring or not, and one would find a lot more confidence among contestants playing a single instance of a 1-in-3 game than one would find amongst contestants playing a 1-in-1b game.

Apart from differing probabilities, how would you explain the higher levels of confidence among players of the 1-in-3 game?

Alternatively, what about a “game” involving 2 electrical wires, one live and one neutral, and a game involving one billion electrical wires, only one of which is live?  Contestants are offered $1m if they are prepared to touch one wire.  No one accepts the dare in the 1-in-2 game but a reasonable percentage accept the dare in the 1-in-1b game.

Is the probable outcome of a single event a factor in the different rates of uptake between the two games?

The reader begs the question by introducing “hot or cold” and “perspiring or not,” which have nothing to do with objective probabilities (long-run frequencies of occurrence) and everything to do with individual attitudes toward risk-taking. That was the point of my original response, and I stand by it. The reader simply tries to evade the point by reading the minds of his hypothetical contestants (“higher levels of confidence among players of the 1-in-3 game”). He fails to address the basic issue, which is whether or not there are single-case probabilities — an issue that I addressed at length in “Some Thoughts…”.

The alternative hypotheticals involving electrical wires are just variants of the original one. They add nothing to the discussion.

*     *     *

Enough of this. If the reader — or anyone else — has some good arguments to make in favor of single-case probabilities, drop me a line. If your thoughts have merit, I may write about them.