Saturday, December 01, 2007

Probability and its relevance to life

Why do we lose money in our ventures, be it business, lottery, gambling, stocks? Marcus Tullius Cicero cited “Probability is the very guide of life.”

Between winning and losing, it is a 50/50 chance of each events happening. Some quarter of the population thinks it is difficult or even impossible to overcome the odds for the market or outcome of an event is always efficient. Warren Buffett said “if the market is always efficient, I’d be a bum with a tin can lying on the street.”

Probabilities of an event happening are like guesses. Whoever that has the most accurate information that goes through a thoughtful process will have a higher likelihood of getting a higher probability of guessing the event rightly. The question in a successful life of getting what you want is “How likely do we believe that some sort of event will occur?” There are again two sides to the guesses, the good guesses and the bad guesses. The theory of probability is a system for making better guesses.

Probability can either be estimated based on its relative frequency (the proportion of times an event happened in the past) or by making an educated guess based on past experiences or whatever important and relevant information or evidence is available.

How likely is it a hurricane strikes Texas?

According to the National Hurricane Center, there’ve been 36 hurricane strikes in Texas from 1900 to 1996. Based on based experience and barring no change in conditions, we can estimate there is about a 37% chance that a hurricane will strike Texas in any given year. This figure is also called the base rate of frequency of outcomes.

In order for relative frequency to be accurate and use it for our future guide, we must ensure that the conditions that produced the past frequency of events remain relatively unchanged.

Besides ensuring conditions remaining pretty much the same as before, we must also look at the variations of outcome and severity – the level of damage caused by an event. Take Tornadoes for example, according to the National Climate Data Center, between 1950 to 1999, there has been an average of 810 tornadoes yearly in the United States. But in 1950, there were 201 tornadoes causing 70 deaths, in 1975 there were 919 causing 60 deaths, and in 1999, there were 1342 tornadoes causing 94 deaths.

At times, a doctor says, “This is the first time I’ve seen this disease. I estimate there’s a 50-50 chance that the patient will come through.” This statement has only two possible outcomes. Either the patient dies or not. Probing further, does then saying 50-50 chance of surviving makes any sense if there is no past date or other evidence to base the probability on? Does it really tell us something? If there is no historical comparable or representative date to base an estimate on, the probability figure only measures the doctor’s belief in the outcome of the event. And to say 50-50 chance of an unprecedented event is a statement any one can intelligently make.

Another doctor says, “According to medical records of similar cases, under the same conditions, 50% of the patients survived five years or longer.” The more representative the background data we have the better our estimate of the probability.

Events may happen with great frequency or rarely. Some events are not repeatable and some events have never happened before. For certain events, past experience may not be representative. Others are characterized by low past frequency and high severity. Unforeseen events occur where our actual exposure is unknown. The more uncertainty there is, the harder it is to find a meaningful probability. Instead our estimate must be constrained to a range of possible outcomes and their probabilities.

Uncertainty increases the difficulty for insurers to appropriately price catastrophes, such as hurricanes or earthquakes. Warren Buffett says:

Catastrophe insurers can’t simply extrapolate past experience. If there is truly “global warming,” for example, the odds would shift, since tiny changes in atmospheric conditions can produce momentous changes in weather patterns. Furthermore, in recent years there has been a mushrooming of population and insured values in U.S. coastal areas that are particularly vulnerable to hurricanes, the number one creator of super-cats. A hurricane that caused X dollars of damage 20 years ago could easily cost 10X now.

But Warren went on to say it is possible to price sensibly.

Even if perfection in assessing risks is unattainable, insurers can underwrite sensibly. After all, you need not know a man’s precise age to know that he is old enough to vote no know his exact weight to recognize his need to diet.

How reliable is past experience for predicting the future then? Peter Bernstein in his book, In Against the Gods, refers to a 1703 letter written by German mathematician Gottfried Wilhelm von Leibniz to the Swiss scientist and mathematician Jacob Bernoulli referring to mortality rates: “New illnesses flood the human race, so that no matter how many experiments you have done on corpes, you have not thereby imposed a limit on the nature of events so that in the future they could not vary.” Even with the best empirical evidence, nobody knows precisely what will happen in the future.

After 9/11, Warren Buffett wrote the importance on focusing on actual exposure and how using past experience sometimes can be dangerous.

In setting prices and also in evaluating aggregation risk, we had either overlooked or dismissed the possibility of large-scale terrorism losses. In pricing property coverages, for example, we had looked to the past and taken into account only costs we might expect to incur from windstorm, fire, explosion and earthquake. But what will be the largest insured property loss in history originated from none of these forces. In short, all of us in the industry made a fundamental underwriting mistake by focusing on experience, rather than exposure, thereby assuming a huge terrorism risk for which we received no premium.

Experience, of course, is a highly useful starting point in underwriting most coverages. For example, it’s important for insurers writing California earthquake policies to know how many quakes in the state during the past century have registered 6.0 or greater on the Richter scale. This information will not tell you the exact probability of a big quake next year, or where in the state it might happen. But the statistic has utility, particularly if you are writing a huge statewide policy.

At certain times, however, using experience as a guide to pricing is not only useless, but actually dangerous. Late in the bull market, for example, large losses from directors and officers liability insurance (D&O) are likely to be relatively rare. When stocks are rising, there’re a scarcity of targets to sue, and both questionable accounting and management chicanery often go undetected. At that juncture, experience on high-limit D&O may look great.

But that’s just when exposure is likely to be exploding, by way of ridiculous public offerings, earnings manipulation, chain-letter-like stock promotions and a potpourri of other unsavory activities. When stocks fall, these sins surface, hammering investors with losses that can run into the hundreds of billions.

Even if we can’t estimate the probability for some events, there may be some evidence telling us if their probabilities are increasing or decreasing. Ask: Do I understand the forces that can cause an event? What are the key factors? Are there more opportunities for the event to happen?

Warren Buffett says on terriorism:

No one knows the probability of a nuclear detonation in a major metropolis area this year….Nor can anyone, with assurance, assess the probability in this year, or another, of deadly biological or chemical agents being introduced simultaneously…into multiple office buildings and manufacturing plant.

Here’s what we do know: a. The probability of such mind-boggling disasters, though likely very low at present, is not zero. b. The probabilities are increasing, in an irregular and immeasurable manner, as knowledge and materials become available to those who wish us ill.

Low frequency events

Adam Smith, a Scottish philosopher, said: The chance of gain is by every man more or less overvalued, and the chance of loss is by most men undervalued.

Supreme Court Justice Oliver Wendell Holmes, Jr. said: “Most people think dramatically, not quantitatively.” We overestimate the frequency of deaths from publicized events like tornadoes, floods, homicides, and underestimate the frequency of deaths from less publicized ones like diabetes, stroke, and stomach cancer. Why? We tend to overestimate how often rare but recent, vivid or highly publicized events happen. The media has an interest in translating the improbable to the believable. There’s a difference between the real risk and the risk that sells papers. A catastrophe like a plane crash makes a compelling news story. Highly emotional events make headlines but are not an indicator of frequency. Consider instead all the times that nothing happens. Most flights are accident-free. Ask: How likely is the event? How serious are the consequences?

John is board a plane tomorrow and wonders, “How likely am I to die on this trip?”

What is the risk of a disaster? First, we need to know the available record of previous flights that can be compared to John’s flight. Assume, we find that in 1 out of 10,000 flights there was an accident. The record also shows that when an accident happens, on average 8 out of 10 are killed, 1 injured, and 1 safe. This means that the chance that a passenger will be involved in an accident is 1 in 10,000; being killed, 1 in 12,500 (10,000/0.8); being injured, 1 in 100,000 (10,000/0.1).

According to the National Transportation Safety Board, the number of passengers killed in air accidents in the U.S. during 1992 to 2001 was 433. For reference, in 2001, the annual number of lives lost in road traffic accidents in the U.S. was 42,119.

That people feel safer driving than flying makes sense since we are orientated towards survival. As Antonio Damasio says in Descartes’ Error, “Planes do crash now and then, and fewer people survive plane crashes than survive car crashes.” Studies also show that we fear harm from what’s unfamiliar much more than mundane hazards and by things we feel we control. We don’t feel in control when we fly.

Why do we lose money gambling? Why do we invest in exotic long shot ventures?

We often overestimate the chance of low probability but high-payoff bets. For example, how likely is it that anyone guesses a number between 1 and 8 million? What is John’s chance of winning “Toto (6 winning numbers out of 45)” if there are 8 million outcomes? What must happen? He must pick 6 numbers out of 45 and if they all match the winning numbers, he wins. What can happen? How many permutations can he chose from? The possible number of ways he can chose 6 numbers out of 49 is 8,145,060. The probability that someone chooses the winning combination is thus one in about 8 million. An odd merely slightly better than throwing heads on 20 successive tosses of a coin.

Imagine the time it takes to put together 8 million combinations. If we assume each combination on average takes a minute to put down on paper, and John spend 24 hours a day writing the numbers, it will take him 15.5 years to write them all down.

Even if John invests $8 million to buy 8 million tickets in the hopes of winning a $15 million jackpot, he may have to share the jackpot with others that picked the winning number. If just one other person picked the winning combination, he would lose $0.5 million.

Why do people play a game when the likelihood of losing is so high? Even if we exclude the amusement factor and the reinforcement from an occasional payoff, it is understandable since they perceive the benefit of being right as huge and the cost of being wrong as low – merely the cost of the ticket or a dollar. It brings us back to Benjamin Franklin’s teaching: “He that waits upon fortune, is never sure of a dinner.”

Chance has no memory

How many times have we heard “My luck is about to change. The trend will reverse.” Sir John Templeton cited: “The four most expensive words in the English language are, ‘This time it is different.’”

We tend to believe that the probability of an independent event is lowered when it has happened recently or that the probability is increased when it hasn’t happened recently. For example, after a run of bad outcomes in independent events that appear randomly, we sometimes believe a good outcome is due. But previous outcomes neither influence nor have any predictive value of future outcomes. There is neither memory nor a sense of justice.

John flipped a coin and got 5 heads in a row. Is a tail due next? It must be, since in the long run heads and tails balance out.

When we say that the probability of tossing tails is 50%, we mean that over a long run of tosses, tails come up half the time. The probability that John flips a head on his next toss is still 50%. The coin has no sense of fairness. As the 19th century French mathematician Joseph Bertrand said: “The coin has neither memory or consciousness.” John committed the gambler’s fallacy. This happens when we believe that when something has continued for a certain period of time, it goes back to its long-term average. This is the same as the roulette player when he bets on red merely because black has come up four times in a row. But black has the same chance as red to come up on the next spin. Each spin, each outcome is independent of the one before. Only in the long run will the ratio of red to black be about equal.

Every single time John tosses, the probability it lands on heads is 50% and tails 50%. Even if we know that the probability is 50%, we can’t predict if a given flip results in a head or tail. We may flip heads ten times in a row or none. The laws of probability don’t count our luck.

John thought, “I got a speeding ticket yesterday, so now I can cross the speed limit again.” Even criminals suffer the gambler’s fallacy. Studies show that repeat criminals expect their chance of getting caught to be reduced after being caught and punished unless they are extremely unlucky.

John finds it comforting knowing it will take another 99 years until the next giant storm happen.

What is a 100-year storm? To predict storms we look at past statistics. We also assume that the same magnitude of storm will occur with the same frequency in the future. A 100-year storm doesn’t mean it happens only once every 100 years. It could happen any year. If we get a once in 100-year storm this year, another big one could happen next year. A 100-year storm only means that there’s a 1% chance that the event will happen in any given year. So even if large storms are rare, they occur at random. The same reasoning is true for floods, tsunamis, or plane crashes. In all independent events that have random components in them, there is no memory of the past.

Controlling chance events

We believe in lucky numbers and we believe we can control the outcome of chance events. But skill or effort doesn’t change the probability of chance events.

Someone offered John to exchange his lottery tickets. John said: “Change tickets. Are you crazy? I would feel awful if my number comes up and I’d traded it away.”

In one experiment a social psychologist found that people were more reluctant to give up a lottery ticket they had chosen themselves, than one selected at random for them. They wanted four times as much money for selling the chosen ones compared to what they wanted for the randomly selected ticket. But in random drawings, it doesn’t make any difference if we choose a ticket or are assigned one. The probability of winning is the same. The lesson is, if you want to sell lottery tickets, let people choose their own numbers instead of randomly drawing them.

The consequences of being wrong

“Pascal’s Wager” is the application by Blaise Pascal of the decision theory for believing in God. Pascal reasoned that it is a better “bet” to believe that God exists than not to believe because the expected value of believing is always greater than the expected value of not believing. He thought: If we believe in God, and God exists, we would gain in afterlife. If we don’t believe in God, and God exists, we will lose in afterlife. Independent of the probabilities of a God, the consequences of not believing are so awful, we should hedge our bet and believe.

Pascal suggests that we are playing a game with 2 choices, believe and not believe: If God exists, and we believe God exists, we are saved. This is good. If we don’t believe, and God is unforgiving, we are damned. If we believe but God doesn’t exist, we miss out on some worldly pleasures. If God doesn’t exist and we don’t believe that God exists, we live a normal life.

Pascal said: “If I lost, I would have lost little. If I won I would have gained eternal life.” Our choice depends on the probabilities, but Pascal assumed that the consequences of being damned as infinite, meaning the expected value of believing is least negative and therefore he reasoned that believing in God is best no matter how low we set the probability of God exists.

John wants to make some extra money and is offered to play Russian roulette. If John wins he gets $10 million. Should he play? There are 6 equally likely possible outcomes when he pulls the trigger – empty, empty, empty, empty, empty, bullet. This makes a probability of 5/6 or 83%.

Should he play this game once? The probability is 83% that he gets $10 million. The probability is only 17% that he loses.

Let’s look at the consequences. If John doesn’t play and there’s a bullet he is glad he didn’t play. If he plays and there is a bullet, he dies. If he doesn’t play and there is no bullet, he loses the pleasure the extra money could have brought him. If he plays and there is not bullet he gains $10 million which would buy him extra pleasure. To play is to risk death in exchange for extra pleasure. There’s an 83% probability that John is right but the consequence of being wrong is fatal. Even if the probabilities favor him, the downside is unbearable. Why should John risk his life? The value of survival is infinite, so the strategy of not playing is best no matter what probability we assign for the existence of “no bullet” or what money is being offered. But there may be exceptions. Someone that is poor, in need of supporting a family who knows he will die of a lethal disease within 3 months might pull the trigger. He could lose 3 months of life, but if he wins, his family will be taken care of after his death.

We should never risk something that we have and need for something that we don’t have and don’t need. But some people pull the trigger anyway. This is what Warren Buffett said about the Long Term Capital Management affair:

Here are 16 extremely bright – and I do mean extremely bright – people at the top of LTCM. The average IQ among their top 16 people would probably be as high or higher than at any other organization you could find. And individually, they had decades of experience – collectively, centuries of experience – in the sort of securities in which LTCM was invested.

Moreover, they had a huge amount of their own money up – and probably a very high percentage of their net worth in almost every case. So here were super-bright, extremely experienced people, operating with their own money. And yet, in effect, on that day in September, they were broke. To me, that’s absolutely fascinating.

In fact, there’s a book with a great title – You only have to get rich once. It’s a great title, but not a very good book. (Walter Guttman wrote it many years ago.) But the title is right: You only have to get rich once.

Why do very bright people risk losing something that’s very important to them to gain something that’s totally unimportant? The added money has no utility whatsoever – and the money that was lost had enormous utility. And on top of that, their reputation gets tarnished and all of that sort of thing. So the gain/loss ration in any real sense is just incredible. Whenever a really bright person who has lost a lot of money goes broke, it’s because of leverage. It’s almost impossible to go broke without borrowed money in the equation.