Lectures


7

LECTURE 9

Our difficulties with probability

Most people buy lottery tickets. Surely, you would think, they understand that whoever is running the lottery is making a profit and that players on average must lose. Do they think about the low probability of winning when they purchase the tickets, one wonders? If not why not? Perhaps they think about the high stakes and ignore everything else. I suspect they do? Why because most people do not feel confortable when it comes to making estimates of the chances of …..

The same happens to many people who make a decision to drive rather than fly, especially right after the news of an airplane crash hits TV. Yet the airline companies and the government agencies put out statistics that tell us over and over again that the probability of an accident per passanger mile is much lower in airplanes than in cars. What is going on? Is it the fact that an airline crash almost always means death and hardly ever in a car crash that influences the decsision rather than the low probabilties. Again there is a lack of comfort when it comes to including events which have low probabilties into decisions.

For example, people who go to Atlantic City believe that if a roulette wheel has stopped at red the last seven times it due to stop at black the next time and they bet on this belief. Mention that the wheel has no memory and they are unfazed. On the other hand if a dice player is coming up with sevens then people bet on the basis of the dice thrower’s “hot” hands. What are your thoughts in a casino?

What is going on? Are we blind to the probabilties of these situations or do we just ignore them? Some psychologists claim that our brains are just not made for taking into probability into consideration when it comes to making decisions. We find it much easier to imagine the consequences of what will happen rather than the probability it will happen. Or failing that we look for correlations in the past between what might happen-the effect, and possible causes. Then we try to dream up a plausible set of connections between the so called cause and the effect. Finally we start gathering together all kinds of information much of which may be different versions of the same thing-what we call redundancy. It is our misconceptions about these elements that attend the judging of rare events that I want to deal with in this lecture. These misconceptions effect how we make decisions where we have to reason on the basis of probabilities. Specifically, I am going to deal with:

A rare event that really happened in Brazil

Here is an interesting true story. It is based on a news report I read about it in the NY Times. Sometime ago, two scrap workers in a small city in Brazil found a piece of metal in an abandoned medical clinic. The piece of metal looked like a big water pipe capped at both ends. The scrap workers had no idea, perhaps, because no one had bothered to label the pipe, or perhaps, someone had scrapped off the label because they did not want to go to the expense of getting rid of it, but inside the pipe was a capsule containing a lethal radioactive material. The pipe was, in fact, part of a containment vessel used to protect the workers in the clinic from the radiation from the capsule inside the pipe. This capsule was taken out and was used in a medical clinic to irradiate people with cancer. The scrap workers thinking the pipe was an ordinary piece of scrap which was not worth salvaging, took the pipe to the city dump and threw it on a trash pile.

There the radioactive pipe was found a few days later by some young boys scrounging in the dump. Somehow, no one has any idea of how, these boys removed the tightly screwed on ends of the pipe and removed a small stainless steel capsule inside. It was this tiny stainless steel capsule that held the dangerous radioactive material, but no with no protection. To these kids, the stainless steel capsule looked like some kind of shiny ornament, so they took it home and showed off their new find to their playmates and to their families. The capsule was handed from one person to another who looked at it admiringly not knowing the danger they faced.

It was only several days later and then, only after these boys who found the capsule and their friends reported that they felt ill, that someone from the city governement came and checked the capsule. They immediately found out that the capsule was highly radioactive. In fact, the shiny capsule contained a radioactive form of Cesium, a highly dangerous radioactive material. In all, 121 people were exposed. Many suffered from radiation sickness with almost certain long term effects on their health. Four died of the radiation.

But the damage was not done. News of the radioactivity in the city spread like wildfire. Before it was over 100,000 people in the city demanded to be checked out for radiation exposure. Not only that, the fear of contamination drove down the price of farm products from the farming areas around this city. Industrial production in the city declined. The story got in the papers throughout Brasil, and visitors from other parts of Brazil avoided coming to this area. A further economic loss. Remember, this is a true story. It is also a rare improbable happening.

Baysian Probability vs. Frequency Probability

Now let us imagine that you find yourself in a decision situation where, as a member of a safety board at a hospital, you are required to help decide on safety measures dealing with radioactive sources kept at the hospital for the treatment of cancer. As part of the decision to examine the various options available to the hospital you have to come up with an estimate of a probability of something like what happened in Brazil taking place again here at your hospital. [ Less you think, this is a far-fetched scenario, let me remind you that about a couple of years ago, some of you may recall, a Brookhaven town official was talking about the possibility of a radiation attack by some locals who had gotten hold of some radium, a very dangerous radioactive material.] Essential to your making a decision is an estimation of the probability that something like happened in Brazil happen in your hospital.

Before you get started, let me note that psychologists tell us that when it come to estimating the probability of a rare or infrequnt event -an event where the potential damage is large, most of us tend to over-estimate the probabilities. We make these over-estimates even though it is clear that the past is a poor predictor of the future when it comes to rare or infrequent events. Probability in these cases can no longer said to be simple statements regarding the relative frequency of certain events like it was in the case of flipping pennies. It is important to understand what we are doing because in most cases where we make hard decisions we are required to estimate the probability of rare or infrequent outcomes.

To get a clue, let us examine what happened in Brazil again. The engineers who designed the radioactive capsule for use at the hospital in Brazil were supposed to have incorporated into their design of the radioactive capsule “the possibility of any concievable contingency”. Think of what that means. It means they were supposed to imagine any number of scenarios of what could happen and to build into the capsule safeguards that made these scenarios extremely improbable. The logic goes like this. If each of the safeguard I put in have a low probability of being overcome, then the chances of a series of events taking place that overwhelm a whole series of safeguards is vanishingly small.

But could the capsule designers ever have imagined that the capsule which came with a hard-to-remove label on the outside in bright yellow saying in large letters "radioactive"would have fallen into the hands of medical clinic owners who anxious to save the cost of having someone remove the capsule would scrape off the radioactive label on purpose and sell the capsule to scrap dealers. Who could have imagined that boys would have ingenuity to break the capsule container apart-a container that was caped with ends that had to be removed with a special tool. Would it occur to these engineers that the shiny stainless steel capsule would be mistaken for an ornament and passed around and admired by the children? These designers made their probability estimates on the basis that an underlying logic is employed by both designers and users. This caused them to underestimate the probability of what happened in Brazil happening. So not just you and I, but individuals knowledgable about the laws of probability have trouble making estimates of rare events.

One of the reasons all of us-trained and untrained people have such a difficult time in estimating the probability of infrequent events lies in the confusion surrounding what we mean by “probable”. There are two basic and distinct schools of thought when it comes to defining what we mean when we talk about probability. The two schools not only disagree, but their disagreement exist at at a definitional level. This means both schools use the same word probability but the word means something quite different to each of these schools. One school calls itself “Frequentists”. The other “Baysian”.

Most of us are familiar with the frequentist view of probability. That is definited as is the relative frequency of events that have already taken place frequently enough for us to have observed a pattern, and, more importantly, are almost sure to take place as frequently in the future. Flipping a penny is an example. To say that the probability of heads turning up on the next flip is 50% means that in the next 100 flips 50 will come up heads and 50 will come up tails. But what does probability mean when we start referring to rare or infrequent events-meaning events which may have never happened before or if so, then infrequently, and are likely to happen in the future only once in a great while. An airplane crash or a reactor blowing up, or a group of boys getting hold of a radioactive capsule.

You and I are less familiar with the baysian schools’s definition of probability. This is also called the subjectivist school. This definition of probability says when we give a number for the probability of an event we are quantifying our beliefs.

To illustrate the two definitions of probability let us conduct a little experiment. I want each of you to think about how you would go about answering this question before you answer the question. The question is, "What is the probability that that more than 10% of this class will end up getting A's?" If you raise your right hand, I will take it to mean that you think there is a high probability that more than 10% of the class will get A's. You can raise your left hand. I will take this to mean that you think the probability is low that more than 10% of the class will get A's. You can choose not raise either hand. I will take this to mean you are willing to say you have no idea, which means you have no basis for making an estimate of this probability.

Now let me change the question to "What is the probability that more than 33% of this class will get F's?" Again you have three choices. yes, no, decline to vote which means you have no basis for venturing a guess.

The frequentists would argue as follows: Let me consider past classes I have completed at Stony Brook. Let me think of what the class grade distribution was in these classes. I will then base my vote on my best estimate of the probability of the outcomes presented to me. In other words the probability is based on a summary of what happened in these other classes. Your decision on how to vote or whether to vote was based on a summary of past behavior of teachers at Stony Brook.

You might argue: I have been in many classes where more than 10% of the students got A's, but I have never been in a class where more than 1/3 of the class flunked. So estimates of probability of future events are based on an estimate of the frequency of similar events that have happened in the past. Probability comes from statistics. Now I want you to imagine yourself as a baysian. You would argue as follows: I have never been in a class of Prof. Nathans, so I have no idea how he grades. Yes, I heard him say something in his first lecture about his intended grade distribution but I have no idea of whether or not this is trustworthy information. Yes, I did talk to a few people who had been in his course in previous years, but I have no idea of how representative their evaluation is. But I do know he is a physicist and physicists are supposed to be tough graders. Finally I have to consider that if he did in fact fail more than 1/3 of any of his previous classes I certainly would have heard about it.

In order to decide how to vote I have to come up with an estimate of the probability of these two outcomes. My estimate of probability will be based on an assessment of my reasonable degrees of belief in these assertions that I have described. Statistics is one of the inputs but not the only one. The statistics influence influences my beliefs. But a great deal else enters. The reasonableness of the outcome. My confidence in what I have heard or read. My own intuition. Your estimates of probability of an outcome is a way of attaching a number to these beliefs. There is a classic book by Ian Hacking called "The Emergency of Probability". According to Hacking. "Probability has two aspects. It is connected with the degree of belief [baysian] warranted by the evidence [in your mind, I might add]. Frequentists would say that we have either no basis or a very weak basis for making a probability estimate dealing with a rare event. According to the frequentists, any attempt to estimate the probability of a rare event compared to trying to estimate a fairly frequent event is like comparing apples and oranges. there is simply no basis for making probability estimates of rare.

So from a frequentist perspective, there is a proper basis to answering that probably more than 10% of the students in this class will get A’s. This is not a rare occurance. There is a statistical basis for estimating a probability. But, frequentists would say that the appropriate answer to my second question about the chances of more than 33% failing is "I decline to answer”. I have no basis for estimating the probability of this outcome since it is a rare event. There are just about no events like this that have taken place, so how can I calculate a frequency distribution.

The baysian approach to estimating probability takes into consideration not only the statistics that says 10% getting A’s happens fairly frequently in classes at Stony Brook and 33% hardly ever happens, but included is your sense of the reasonableness of these outcomes. It is quite reasonable to think that someone would give a class in a subject like this where the criteria for grading must necessarily be so imprecise and come out with 10% A’s. At the same time, it is unreasonable to think of an outcome in which 33% failed. In other words you estimate of the probability of these two outcomes represents in summary form an assessment of everything you know [ and, if you want to include your subjective views, your judgements of me, others you have known who took the course, what you have heard so far, etc]. So if I asked you why you raised or did not raise your hand in the either case, you have to provide me with a reasonable statement. Clearly the baysian school incorporates more subjectivity into its estimates. And this is why there is so much controversy that surrounds it.

This historic duality in defining probability is, perhaps more than anything else, responsible for creating our confusion about probability. We are never sure about whether someone is making their estimate from a purely statistical [penny flipping] frequentist perspective or giving us their baysian assessment. When it comes to your including experts judgements into your decision making you have to find out which meaning is intended when you ask such questions as, "What are my chances of ............? You can fill in the blanks. In medicine it may be "having such-and-such illness". In the case of genetics it may be "having a genes which gives me a pre-disposition for ....." In finance it may be "losing more than 50% of my money over the next x months" Both can be included in your making a hard decision, but it is important that you know the basis [frequentist or bayesian] on which the answer is based. Clearly there is uncertainty involved in either basis of estimating probabilties. If your decision depends on how well you know these probability estimates, the basis on which these estimates are made is a starting point for determining how large the uncertainty is. (Lawyers say they believe that each case is based on its own individual merits; they favor bayesian ideas of probability. In reality they are well aware of statistics playing a role in the outcome. That is why they hire statistically trained people to help them in jury selection.)

The hot hands effect: a made-up cause to explain an illusory effect

I recall when I was working on Wall St. I used to hear traders making such nonsense comments as " how come every time you see the stock market go up, the President is traveling overseas." I don’t even know if that statement is true, but it is clear to anyone that follows the stock market gurus that they all spend lots of time searching for plausible causes to explain what we think is a correlation between this event and that condition. Wall St. is not the only place there this endless speculation goes on. Study the paper, listen to TV, and you will hear this speculation over and over again. Here is the pattern it follows.

The following should be very familiar to you. Some group does a statistical study. They find a correlation between age and beauty, height and weight, staying up late and grades on finals. Pick whatever suits you. There are statistical studies on just about everything these days. The results may or may not be sound statistically. It really does not matter. If the results are just a little bit different from what might have been expected, then the next step is to look for a “reasonable” cause to explain the result. How come, I have always wondered that some people never seem to stop searching for a causal explanation for every kind of imaginable correlation-even ones that may not be real. Reasonable is of course a relative word. What is reasonable to one person may not be to another.

One of the strangest of these made-up causal models is the so called "hot hands" effect. In basketball, the hot hands effect refers to the positive feedback governing players which makes them think that they are more likely to score after scoring and to miss after missing. The same applies to stock market investors. A careful statistical analysis that has been done over and over shows that the hot hand effect is just not real. Statistically, it is a fiction. Statistically, a success following a failure is just as probable as a success following a success or a failure failing a failure. In other words, there is no correlation between past and future performance. It is pure illusion. Yet players and of course, investors insist the hot hand effect is real. I found this hot hand effect to be especially pernicious on Wall St. Yes, I even admit to occasionally falling for it myself. Tversky concludes, "The belief in the hot hand [effect] and the detection of streaks in what is a random sequence is attributable to the general misconception about chance." Why is this? We know that all things being equal in this world, if we observe games of chance like crap shooting with fair dies, than the most probable outcome of the game is random and therefore hot hand streaks are to be expected. Yet we as humans apparently have a psychological need to see a pattern in this; a correlation between past and future events. But to bolster our belief in this illusion, we need a plausible cause of why the past performance should be tied up with future performance. In the case of the basket ball player, he tells himself that he every time he gets a good nights sleep last night he plays better. The gambler clutches at causes like the way he holds the dies just before the toss, or the presence of a beautiful blond near him.

More seriously, does having four healthy weeks in a row indicate the success of a medical treatment? Do four bad weeks indicate a failure? No, no more than four heads in a row within a sequence of coin tosses indicates that somebody messes with the coin.

This is not a trivial point. Many mutual funds rely on computer models that can simulate market behavior. The trouble is that when it comes to checking out computer simulation models by comparing their results with actual behavior in the past [this is one of the primary test for deciding whether or not they will be useful for prediction of what is going to happen in the future], we have to be on our guard we do not get tricked by the hot hand effect. How do you do this? You use a computer to "run the model" every day for the last five years and trade whenever the model tells you to. You then keep track of the profits and losses day by day. This is called back-simulation. We then see if we would have made money or lost money over this 5 year period. If we do, we think we have a pretty good predictive model. I used this technique over and over again.

Is there anything wrong with back stimulation? Yes, it suffers from the hot hand syndrome. Just as in the hot hands effect, you are lulled into believing that during long periods when the model is making money month after month, the model is working well, but during periods of loss, you tend to rationalize that market conditions have changed. When and if you become investors in the market you will find this kind of argument used often by investment advisors. So take heed and remember the "hot hand effect".

Every correlation does not have a cause

I have noted that estimating the probability of possible scenarios as well as the risks and benfits associated with these scenarios is an essential part of making decsions. These estimates rely on an knowledge base derived from statistical inferences as inferences that follow from what we know about what caused what. In actual practice when making these estimates we tend to go back and force between the statistics and the causal models rather than combining them in any systematic manner. We use what we know about cause and effect to reinforce the statistics and visa versa. That is fine, but we do more than this.

We so believe in this mutual reinforcement between cause and effect and statistical correlation, we can be persuaded to revise the estimate of the probability of some event taking place, if someone has provided them with a plausible model of cause and effect which supports their probability estimate.The point is that we can easily be tricked by observing what we believe to be a statistical correlation and then trying to increase our confidence in the outcome by saying the results are reasonable based on the existence of a plausible causal model. It is done all the time and by experts who should know better. Draw a statistical sample from a small number of cases. Observe what looks like a statistical correlation. try to justify it by developing a plauible cause effect mechanism that this result would be consistent with. But this type of reasoning can lead to erroneous decisons.

Most frequently, the cause-effect model should comes first and should predict the correlation as a consequence. That is a much stronger argument. Yet , you will find these articles in the paper every day. Ex post facto is the name we give to them. Arguments after the fact. They are totally unjustified. Yet such spurious type of arguments appear in the financial section of the NY Times as well as the Wall St. Journal all the time. They also appear in stories in the paper on health. We first find out this or that medicine works and then we come up with some kind of ad hoc causal model to explain our statistical results.
It may look different but is it really a different version of the same thing.

Suppose I want to confirm a diagnosis by a physician by getting a second opinion. I have available two methods. One allows me to ask the first physician to recommend someone in his office. He talks to that physician and tells him what he has found out. I then go and see this other physician. In this case we say there is a chance that the two opinions are redundant. The second method allows me to locate a physician for a second opinion by opening up the phone book, closing my eyes and picking a physician by putting my finger on a name. The first method runs the risk that the information I get in the second opinion will be biased by what I heard from the first physician. The second has no chance of this, because we have insured that the two opinions will be independent of one another by picking them out randomly and seeing to it that there is no exchange of information.

One definition of redundancy is needlessly repetitive. Another is duplicative. The reason that we have to deal with redundancy is that we often get into a situation where we hear confirmation of a piece of information from more than one source. We run the danger of interpreting what is essentially the same thing from two sources as offering reduced uncertainty. This in turns increases ourc confidence in the outcome. But this increased confidence is spurious if in fact the two sources are redundant.

You will often hear someone say by way of arguing the validity of some piece of information, “But the chances of both of those particular results coming from two different sources is vanishingly small.” All of which is true if indeed the two sources are independent. But if they are not then the chances may not be all that small.

This review on the psychological factors that influence decisions made under uncertainty are really just a quick review of what is a very rich and diverse source of material. Tversky and Kahlman’s orgininal work lead to the birth of a new field call behavioral economics. Particularly as it relates to decision making, behavioral economics offers us new incites into how people go about making decisions.

I first came across this field when I was working on Wall St as part of a team putting together computer based models for investing in the stock market. I found it then to be a valuable source of ideas not just for explaining market behavior, but of predicting future behavior. So the field is not just intellectually interesting, it has a number of practical ramifications as well not just in financial decisions but in all kinds of decisions.

Summary and Comment

This is the end of the lectures having to do with psychology. I should think by now that you must be confused. On the one hand you are being told how to make decisions in a rational objective manner, and on the other that we as humans are subject to all kinds of errors and misconceptions. It is confusing. But if out of the confusion comes an informed skepticism about what we are told, and how we should behave when it comes to making hard decisions, then I consider that the course will have been a success.