Probabilities and Number of Possible Outcomes

PROBABILITIES AND NUMBER OF POSSIBLE OUTCOMES

Probability is the very guide of life. — Marcus Tullius Cicero

How likely do we believe it is that some event will occur? Probabilities are like guesses. But as Richard Feynman said in his Caltech Lectures on Physics: "There are good guesses and there are bad guesses. The theory of probability is a system for making better guesses."

We can either estimate the probability based on its relative frequency (proportion of times the event happened in similar situations in the past) or we can make an educated guess using past experiences or whatever important and relevant information and evidence that is available.

We can also count possible outcomes. When all possible outcomes are known and equally likely, we can calculate a probability by dividing the number of favorable outcomes by the total. Fair coins and dice provide familiar examples. We can also calculate probabilities for unequal outcomes if their probabilities are known; merely counting them is then insufficient. However we use the notion of probability, we need to follow its basic rules.

How likely is it that a hurricane strikes Texas?

According to the National Hurricane Center, there have been 36 hurricane strikes in Texas from 1900 to 1996. The calculation of about 37% (36/97) divides recorded strikes by years observed. The ratio—36/97—gives an average annual strike count. It is not automatically the probability of at least one strike in a year: several strikes could fall in the same year. Using either measure to forecast also requires considering changes in conditions.

We must make sure that the conditions that produced the relative frequency can be expected to be pretty much the same before we can use it as a guide for the future. We must also look at variations of outcome and severity (how much damage an event may cause). Take tornadoes as an example. According to the National Climatic Data Center, from 1950 to 1999 there was an average of 810 recorded tornadoes yearly in the U.S. But in 1950 there were 201 recorded tornadoes (causing 70 deaths), in 1975, 919 (causing 60 deaths), and in 1999, 1,342 tornadoes (causing 94 deaths).

Doctor's estimates

A doctor says, "This is the first time I've seen this disease. I estimate there is a 50-50 chance that the patient will survive." This statement has only two possible outcomes. Either the patient dies or not. Does it really make any sense to say "a 50-50 chance" if there are no past data or other evidence to base the probability on? Does it really tell us something? If there are no comparable data or other supporting evidence, the figure expresses the doctor’s belief without showing how well it is supported.

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 background data or evidence we have the better our estimate of the probability.

To narrow down the probability figure even more, we need a relevant comparison group, i.e., a group to which a frequency refers. In the hurricane example, we defined the probability for a specific comparison group, referring to recorded hurricane strikes (36 times in Texas during the 97 years for which we have data). To estimate the chance of a year with a strike, we would need the count of affected years.

Uncertainty and exposure

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 (measures vulnerability and potential cost or loss) is unknown. Sometimes people react to an event by avoiding or preventing it in the future causing a change of the event's future probability. Other times one bad event may increase the chance of another. For example, an earthquake may cause landslides, floods, or power blackouts. The more uncertainty there is, the harder it is to find a meaningful probability number. It may be more useful to examine plausible outcomes, ranges of probabilities, and the limits of what we know.

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.

Occasionally, also, the unthinkable happens. Who would have guessed, for example, that a major earthquake could occur in Charleston, S.C.? (It struck in 1886, registered an estimated 6.6 on the Richter scale, and caused 60 deaths.)

But it may still be possible to price sensibly. Warren Buffett says:

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 nor know his exact weight to recognize his need to diet.

Warren Buffett also considers a worst case scenario:

Given the risks we accept, Ajit [Ajit Jain; manager of Berkshire's reinsurance operations] and I constantly focus on our "worst case," knowing, of course, that it is difficult to judge what this is, since you could conceivably have a Long Island hurricane, a California earthquake, and Super Cat X all in the same year. Additionally, insurance losses could be accompanied by non-insurance troubles. For example, were we to have super-cat losses from a large Southern California earthquake, they might well be accompanied by a major drop in the value of our holdings in See's, Wells Fargo and Freddie Mac… We do, though, monitor our aggregate exposure in order to keep our "worst case" at a level that leaves us comfortable.

How reliable is past experience?

In Against the Gods, Peter Bernstein 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 corpses, 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 the September 11, 2001, catastrophe, Warren Buffett wrote on the importance of focusing on actual exposure and how using past experience sometimes may 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 (after adding related business interruption claims) 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 a 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 are 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 for some events can't estimate their probability, there may be some evidence telling us if their probabilities are increasing or decreasing. Ask: Do I understand the forces that can cause the event? What are the key factors? Are there more opportunities for the event to happen?

Writing about terrorism in that period, Warren Buffett said:

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 plants… 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.

Number of possible outcomes

Toss a coin once. There are 2 possible outcomes. Roll a die once. There are 6 possible outcomes. These outcomes are equally likely if the coin and die are fair. The repeated rolls below also assume independence. Roll a die twice. What can happen? There are 6 possible outcomes on each roll and therefore 36 possible combinations or outcomes when rolling a die twice. Roll a die 3 times. There are 216 possible outcomes.

With equally likely outcomes, a single specified outcome becomes less likely as the number of possible outcomes grows. For example, only one face satisfies "roll a die once and observe a six", while there are 6 possible faces. An event can include several outcomes, so its probability depends on how much of the full set it covers. If outcomes are not equally likely, we must use their probabilities rather than just count them.

Treat rolling a die 3 times as 3 separate events where each event is "observe a six". What we see from the above is that the more events that must happen to achieve some scenario or wanted outcome ("3 sixes in a row"), the less likely the scenario is to happen. Observing "anything but 3 sixes in a row" is an unwanted event. There are 215 outcomes or ways for this unwanted event to happen out of 216. This means that it is very likely that the unwanted event happens.

We talk about what is likely to happen in the long run. We might be lucky and roll 3 sixes in a row. We must also consider the consequences of an unwanted outcome.

If the elementary outcomes are equally likely, more bad outcomes than good ones means a bad result is more probable. In a real system, however, failure paths can differ greatly in likelihood. Counting them does not by itself establish the chance of failure.

It means that surprises, coincidences, rare events and accidents happen, somewhere, sometime, and to someone if they have opportunities to happen.

Where a serious risk can reasonably be avoided, eliminating the exposure may be more useful than trying to estimate its probability precisely. For example, we can reduce risk by increasing the number of wanted possible outcomes, reducing the number of unwanted possible outcomes, reducing the magnitude of consequences or avoiding certain situations.

Ask some relevant questions

• Event: Type of event? Frequent? Unique?
• Causes: What can initiate and cause the event? What factors contribute? What conditions and circumstances must be present? Have the causes changed over time?
• Exposure: Known? Measurable? Possible consequences? Magnitude of consequences/loss? What's the worst that can happen?
• Probability: Distribution of possible outcomes over time? Stable? Relative frequency or relevant past experience? Number of observations? Relative likelihood of different size of losses? How is average frequency produced? Variability in outcome and severity? Dependence on human factors?
• Representative: Past data representative or change in conditions? Evidence of changes in causes or frequency of event? Temporary or permanent change? Small sample or too short observation time? Changing exposure as time proceeds?
• Backups: Backup failure rate?

Low frequency events

The chance of gain is by every man more or less overvalued, and the chance of loss is by most men undervalued — Adam Smith

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 and 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 is 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 boarding the day flight from Los Angeles to Washington 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. 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 one 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); and being injured, 1 in 100,000 (10,000/0.1).

According to the Federal Aviation Administration, Dr. Arnold Barnett of Massachusetts Institute of Technology (MIT), a widely recognized expert on air traffic safety, measured a passenger's odds of surviving the next flight. It related the probability of not being in a fatal air carrier accident and the probability of not surviving if a fatal accident happens. In the year 2000 the odds were 5.8 million to 1.

According to the National Transportation Safety Board the number of passengers killed in air accidents in the U.S. during 1992 through 2001 was 433 (including the 232 aboard the four hijacked flights on September 11, 2001). 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 oriented 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 14 million? What is Mary's chance of winning "Lotto 6/49" if there are 14 million outcomes? What must happen? She must pick 6 numbers out of 49 and if they all match she wins. What can happen? How many numbers can she choose from? The possible number of ways she can choose 6 numbers out of 49 is 13,983,816. The probability that a single entry matches the winning combination is therefore one in about 14 million. Merely slightly better than throwing heads on 24 successive tosses of a coin.

Imagine the time it takes to put together 14 million combinations. If we assume every combination on average takes 1 minute to put down on paper, and Mary picks numbers for 24 hours a day, it will take her 27 years to write them all down.

Even if Mary invests $14 million to buy 14 million tickets in the hopes of winning a $20 million jackpot, she may have to share the jackpot with others that picked the winning number. If just one other person picked the winning combination, she would lose $4 million (20/2 - 14).

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 pay off, 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. Remember the advice of Benjamin Franklin: "He that waits upon fortune, is never sure of a dinner."

Mathematical expectation

A lottery has 100 tickets. Each ticket costs $10. The cash prize is $500. Is it worthwhile for Mary to buy a lottery ticket?

The expected net value is the probability of winning (1 in 100) multiplied by the net gain—the prize ($500) less the ticket cost—minus the probability of losing (99 out of 100) multiplied by the cost of playing ($10). For each outcome we take the probability and multiply the consequence (a reward or a cost) and then add the figures. This means that Mary's expected value of buying a lottery ticket is a loss of about $5 (0.01 × $490 - 0.99 × $10).

We need to distinguish the possible result of a single game from the average over many repetitions. Expected value is a probability-weighted average. Under suitable conditions, the average result of many comparable bets tends toward that value. Expected value tells Mary that she on average should expect to lose $5 every time she plays if she plays the same lottery over and over. Not what she can expect from a single game. Mary has a 1% chance of winning the lottery and if she wins, her gain is $490. She has a 99% chance of losing $10.

Most of our decisions in everyday life are one-time bets. Choices we face only once. Still, this is not the last decision we make. There are a large number of uncertain decisions we make over a lifetime. We make bets every day. Expected value can help compare these decisions when probabilities and consequences can be estimated. It is only part of the choice: a favorable average cannot make an unaffordable loss acceptable or guarantee a good result.

John placed $38 on the roulette table.

The house edge makes ordinary roulette bets unfavorable over repeated play; a player can still finish ahead over a particular period. If we play at a casino once we may be lucky and in the short run win some money, but we should expect to lose in the long run. The casino has the advantage.

There are 38 different numbers (including double zero) on the roulette wheel. When the croupier spins the wheel there is an equal chance of the ball landing in any one of the 38 slots. John puts $1 on a single number. If his number comes up, he wins $35. On average his expected value of a one dollar bet is a loss of 5.26 cents (1/38 × $35 - 37/38 × $1). Over the long run John loses an average of 5.26 cents for every dollar he places on the table. The odds are set so they average out in the casino's favor.

"If I just stay at the table long enough, the odds turn in my favor and I will win back everything I lost." But this is what the casino wants us to believe. The casino can't predict the outcome of any particular bet but over many bets the house edge gives the casino a positive expected profit, though realized profit is not guaranteed. As a casino operator said: "What I love is the risk. Some nights we make money, and other nights we make more money."

Continuing to gamble after a win can expose those gains to further losses. Henry Howard Harper offered the following historical account in The Psychology of Speculation. Its claim about profit without a house edge should be read as his assertion, not as a consequence of the expected-value calculation above:

It is said to be a proven fact that the chances are so much against the player, that a roulette wheel can be run at a profit, even if the percentage in favor of the house is entirely eliminated. This is due to the fact that the excitement of play causes a certain confusion of mind, and players are prone to do the wrong thing; for instance, double their bets when in an adverse run of luck and "pinch" them when luck is running favorably. Or, on the other hand, players who have pressed their advantage and doubled in a run of favorable luck will continue stubbornly to plunge long after their luck has changed. Precisely the same psychology applies to trading in stocks.

Chance has no memory

"My luck is about to change. The trend will reverse."

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. For independent trials with known, fixed probabilities, previous outcomes do not change the probabilities of the next trial. There is neither memory nor a sense of justice.

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

For a fair coin, a tails probability of 50% means the long-run proportion tends toward half; it does not promise exact balance. The probability of heads on Mary’s next independent 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." Mary 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. Over many independent spins of a fair wheel, the proportions of red and black tend toward equality; no particular stretch must balance out.

Every single time Mary plays, the probability it lands on heads is 50% and lands on 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 out luck.

"I got a speeding ticket yesterday, so now I can cross the speed limit again," said John. 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.

Mary finds it comforting knowing it will take another 99 years until the next giant storm happens.

What is a "100-year storm"? To predict storms we look at past statistics i.e., how often in the recorded past a storm of a certain magnitude has occurred. 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 event only means that there is a 1% chance that the event will happen in any given year. The waiting-time point assumes stable annual risk and independence. The same logic applies to other events only when those assumptions fit; weather patterns, damage, and changes made after a disaster can alter future risk.

Controlling chance events

The craps table was filled with people tossing their die soft and asking for a low number.

We believe in lucky numbers and we believe we can control the outcome of chance events. In a fair random drawing, choosing a number personally or concentrating on the result does not change its probability.

"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.

Gains, losses and utility

The 18th Century Swiss mathematician Daniel Bernoulli said: "A gain of one thousand ducats is more significant to a pauper than to a rich man though both gain the same amount." This means that the utility or personal value of an outcome differs for different people and at different stages in life. For example, our preferences change as our state of wealth changes.

Often when we make financial decisions we don't consider our total wealth. Instead we judge a decision by evaluating changes measured in terms of short term gains and losses.

"Should I invest?" "There is a 50% probability that I gain $10,000. There is a 50% probability that I lose $4,000." "Since I get pleasure from gains and the expected value ($3,000) is positive, I decide to invest."

To use expected utility, consider what each outcome would mean in the context of total wealth and personal preferences. The reply below assumes a person whose aversion to the possible loss outweighs the attraction of the gain; wealth alone does not determine that judgment.

"My present wealth is $1,000,000. Do I choose $1,000,000 for sure or $996,000 or $1,010,000 with equal probabilities?" "Since the expected utility resulting from integrating the favorable investment with my wealth is lower than the expected utility of my current wealth, I don't invest."

Remember that the concept of utility is about the personal value of potential consequences and is therefore different for different individuals. An individual who finds that the expected utility resulting from integrating the mentioned investment with his wealth are higher or about the same as the expected utility of his current wealth, will choose the favorable investment.

Ask: What do I end up with? How much will I have if I succeed and how much will I have if I fail? How certain am I? What is the expected utility?

The consequences of low frequency events

Compare two possible losses, each with no loss if the listed event does not occur:

Outcome	Probability	Cost of consequence	Expected value
A	10%	-90	-9
B	90%	-10	-9

These separate loss scenarios have the same expected cost, but the losses, when they occur, differ greatly in size. We can't only look at how likely an unwanted event is to happen. We must also rate the magnitude of its consequences. Before taking an action, ask: What are the benefits and costs? What might go wrong? How can it go wrong? How much can I lose? What is the probability and consequence of failure over time? How can the probability and consequence of failure be reduced?

What if the probability of success is high but the consequence of failure is terrible?

The consequences of being wrong

Take no chance whatsoever with food poisoning, and stay away from places where others have been recently killed, regardless of what the mathematical laws of probability tell you. — Edward Wilson (Professor Emeritus, from Consilience)

"Pascal's Wager" is Blaise Pascal's argument for believing in God. Pascal reasoned as follows: 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, with the following consequences:

Probability (p)	Believe	Don't believe
God does exist (p)	Saved (good)	Damned (bad)
God does not exist (1-p)	Inconvenience	Normal life

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.

Expected value of believing = p (the value of being saved) + (1-p)(the cost of inconvenience) Expected value of not believing = p (the cost of being damned) + (1-p)(the value of living 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 is infinite, meaning the expected value of believing is least negative and therefore he favored belief for any positive probability within that setup. The argument depends on its assumptions about the available beliefs, their consequences, and the use of infinite values.

John wants to make 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 the probability 5/6 or 83%. This is the same as saying that John is playing a lottery with only 6 tickets where one ticket is lethal.

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 was 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 was no bullet he loses the pleasure which the extra money could have bought him. If he plays and there is no bullet he gains $10 million which would buy him extra pleasure. To play is to risk death in exchange for extra pleasure. There is 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 we have and need for something we don't need. But some people pull the trigger anyway. This is what Warren Buffett said about the Long-Term Capital Management affair:

Here were 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 ratio in any real sense is just incredible… Whenever a really bright person who has a lot of money goes broke, it's because of leverage… It's almost impossible to go broke without borrowed money being in the equation.