Rolling the Dice With the Bernoulli Brothers
Rolling dice with the Bernoulli Brothers …
“Decisions are the choices we make when the outcome is uncertain.”
Risk is a crazy business. But it is understandable. Even quantifiable. Mostly.
Game theory, investment strategy, actuarial science, military strategy, weather forecasting, economic prediction, and the mundane choices we make every day – which route to take to work, which checkout line at the grocery, what to have for breakfast, with whom to flirt – are all exercises in risk. So, also, is the way we fly. Risk is where the laws of probability meet the consequences of our actions.
Some days I think I am teaching people to fly airplanes. Every day I know my job is to teach people to evaluate and mitigate risk. Understanding risk is not something we do well. People who watch a lot of television think violent crime is out of control, when, in fact, it has diminished steadily for decades. Most of us who follow world affairs believe that the world is awash in war, whereas the loss of life in war worldwide is much less than in past decades. (See Steven Pinker, The Better Angels of our Nature: Why Violence Has Declined. And see these graphs.) A coin comes up heads 50 consecutive times. Some of us believe intuitively that the odds have now become greater that the next flip will be tails. One in a million of us will win a lottery; one in four of us will be involved in an auto accident, yet we believe – and drive as if – the accident will never happen to us. Parents who live in fear of a school shooting refuse to vaccinate their children against life-threatening disease. And most of us who fly airplanes seldom practice emergency procedures because emergencies happen to other pilots not so virtuous as us.
We pilots are similarly lousy at estimated risk. 
We worry what will happen to us if the engine stops, when the data shows clearly that this is not how we will come to grief in an airplane. Sudden, unforeseeable, act-of-God, total engine failures in a well-maintained airplane figure in a vanishingly small percentage of serious flying accidents. We have long known the choices that lead to grim outcomes in flying: running out of fuel; flying into weather for which either the pilot or the airplane is ill-equipped; and low-level maneuvering. These are not acts of God. These are deliberate decisions. These are actions that we choose. Why pilots keep making these choices is the baffling part. The calculations are simple; the choices are apparently harder than we suppose. We are talking here of risk. I have been thinking lately of the history of risk and how we have spent centuries trying to understand its nature and quantify it.
We have talked elsewhere (Wisdom is Where You Find It) about how to emotionally and conceptually handle emergencies. Here we want to talk about the mathematics and psychology of risk as it has come to be understood through the centuries. How we balance risk and reward. How we hope to make choices that avoid emergencies. How we make decisions when the outcome is uncertain.
Thinkers have grappled with this question from the early days of civilization. The Greeks, the Romans, and the civilizations of Asia all explored the question. The first element of risk analysis –- probability – required the mathematics that made quantification of probability possible.
Once the invention of zero and the other foundational ideas of mathematics fell into place, the mathematics of probability began to take shape, first in the heart of the gambler. The gambler who understands that, given an infinite number of tosses, the dice will inevitably produce a given number a certain percentage of the time has a mathematical advantage – and, if he can finance a sufficient number of tosses, an unbeatable strategy.
Galileo produced an essay translated as “On Playing Dice” in the early seventeenth century in which he addressed the mathematics of various combinations in the throwing of dice.
In the 17th century, a trio of brilliant French mathematicians began to debate the odds of certain occurrences in games of chance. Blaise Pascal, Pierre de Fermat and Antoine Gombaud (known as the Chevalier de Mere) were puzzled by such questions as this: Which is more likely? Rolling a 6 on four throws of a dice or rolling a “double six” on 24 throws with two dice? Of such questions were early calculations of probability founded. (The answer, by the way, is one six in four throws by a narrow margin of approximately. .51 to .49. Narrow to be sure but mathematically certain.)
Fermat, a lawyer by trade but a man of seemingly impossible erudition, in his correspondence with Pascal, made major contributions to the unwinding of probability, but made a more lasting contribution in number theory and left the world an enduring mathematical puzzle and mystery in “Fermat’s Last Theorem.” The chevalier was primarily interested in leveraging the inchoate math of probability to his advantage at the gambling tables. Eventually, probability came to be defined by three axioms: 1) the probability of all occurrences is 1, 2) probability has a value which is greater than or equal to zero, and 3) when occurrences cannot coincide their probabilities can be added. As mathematics ventures beyond the concrete, finite world of the six-sided die and the coin flip and enters the natural world with its vast array of bewildering, interactive forces, the mathematics becomes complex sometimes beyond our computational ability, but still, nevertheless, obeys known laws. This is the world in which we live and fly.
By the end of the seventeenth century, the mathematics of probability was well established, in part a product of the explosion of thought and innovation and experimentation set in motion by the Renaissance. With the tools of quantification in hand, thought turned to the more nuanced question of the human element, the possible outcomes of risk and how we as individuals assess the value or the cost of those outcomes.
Once probability was calculable, the second element of risk, the nature and magnitude of outcome, began to arise. The modern concept of risk and decision theory evolved in the 17th century. Theologians and gamblers, the sacred and the profane, were the first to grapple with the question. One might argue that those two pursuits are among those with the highest risks, one existential and the other financial. For the theologian, these choices offer the chance of existential reward or eternal loss, as in heaven or hell. The concept of probability was of particular interest to men who gambled as the most clever of men came to understand that, in games of chance, the gambler who understands the odds has the advantage. The question of whether to gamble, weighing the outcome along with the chance was considered only later. And for the theologians, the choice was posed as a risk of salvation or damnation. For the gambler, profit or loss. High stakes inspired careful thought among the practitioners of theology and gambling.
Blaise Pascal, the French mathematician, Catholic theologian, and polymath, posed the question whether to believe in God or not to believe in God is the better bet. Granting that belief is not a conscious choice and the question of God’s existence cannot be answered rationally, Pascal assumed the probability of God’s existence to be 50-50 and he then turned to the possible outcomes. If we believe that God exists and we are correct, we earn salvation, while there is no penalty should we be wrong. (The jester might suggest that the penalty was the devout foregoing of libertine pleasure, which might have entered into Pascal’s personal interests, if not his calculations. He was, at times in his life, a habitué of the gambling dens and, at other times, attracted to an ascetic religious fanaticism so ascetic that he abandoned not only earthly pleasures but mathematics.) To deny that God exists and act thusly risks eternal damnation should we be wrong. To believe in God and to act thusly seems the better road. Here is one of the first analyses of risk that takes into account not only probability but outcome and its magnitude, which we now refer to as value, or, more uniquely, utility. It is the consideration of consequence that is the second, equally important, component of the risk equation.
In 1662, the Port-Royal-des-Champs abbey near Paris published a series of volumes known as Logic, or the Art of Thinking, a remarkable work still available today from the Cambridge University Press, usually attributed to Antoine Arnauld and Pierre Nicole. Although largely a theological text, it included profound chapters on probability, including the statement, “Fear of harm ought to be proportional not merely to the gravity of the harm, but also to the probability of the event.” Here is the first export of Pascal’s dilemma into the secular world of decision making, albeit in the obverse. Pascal assumed the probability to be fixed and weighed the potential outcomes. The writers of the abbey stipulated to the outcomes and reminded us to factor in the probability. Together, we have objective probability and subjective consequence. We are beginning to put together the risk equation.
The authors of the Port-Royal Logic framed the fundamental concept of risk when they wrote, “Fear of harm ought to be proportional not merely to the gravity of the harm, but also to the probability of the event.” Read that again. This is the risk matrix we know today and which today seems intuitive, but its development was one of the great achievements of early mathematics and philosophy.
In 1731, a paper presented to the Imperial Academy of Sciences in St. Petersburg, the Specimen Theoriae Novae de Mensura Sortis (Exposition of a New Theory on the Measurement of Risk) argued that “the value of an item must not be based on its price, but rather on the utility that it yields.” The author of the Port Royal Logic argues that only the pathologically risk-averse make choices based on the consequences without regard for the probability. An unpleasant consequence may be worth the risk if the probability is unlikely. The New Theory argues that only the foolhardy considers only the probability while ignoring the consequences. Even an unlikely consequence is too risky if it is truly dire.
The author of the New Theory of 1731, first presented to the Academy in 1731 and later published in 1738, is none other than Daniel Bernoulli, he of the Bernoulli principle, which we are told makes airplanes fly.
The idea that a fluid’s pressure lowers as its flow accelerates, and vice versa, Bernoulli’s principle from his work on fluid dynamics, the Hydrodynamica of 1738, certainly plays a role in the generation of lift, although aerodynamicists seem still to be arguing about the extent of its contribution. Today we are interested in another aspect of Daniel Bernoulli’s intellectual curiosity and genius, his study of risk. Daniel Bernoulli was then 38 years old. He was one of a remarkable family of 17th and 18th century Swiss mathematicians. The brothers Jacob and Johann were each brilliant mathematicians. Among many other advances in theoretical mathematics, Johann introduced the concept that 0 divided by 0 was an indeterminate number, any number. The problem of what to do with zero had troubled mathematicians since the 7th century. And Jacob discovered the approximate value of the irrational number “e” in his exploration of compound interest, explored the mathematics of the binomial trial, which came to be known as the Bernoulli Trial, a trial with only two possible outcomes of unchanging likelihood and made many other contributions to mathematics, including the first use of the word “integral” to refer to the area under a curve, techniques for solving separable differential equations, and the invention of polar coordinates. His major contributions to probability and expected values were published posthumously in “The Art of Conjecture” in 1713. Johann’s sons, Nicolaus, Johann II, and particularly Daniel, were responsible for fundamental discoveries in calculus, number theory, statistics, and probability. Daniel’s paper is one of the most profound explorations of the complicated relationship between the objective measurement of probability and the instinctive, often subjective, sense of consequences, the balance between what we can know and what we can feel, between likelihood and its impact on our lives, between what the mathematics tells us we should choose and what our heart drives us toward.
I find it fascinating that Daniel Bernoulli, who gave us the principle known to every student of the wing, also gave us some of the earliest work on the calculation of risk. There you have the two challenges given to we aviators – how to fly the contraption and how to keep it from killing us.
Taken together, we have for the first time the two elements of risk assessment: probability, which can be an objective measure – every flip of the coin is a 50-50 probability of heads or tails — and consequence, whether reward or loss, which is a subjective judgment. Objective and subjective. Calculation of the objective element has been the fundamental problem in the evolving mathematics of probability. The mathematics can be complicated, but are at least theoretically computable. Evaluation of the subjective consequence has been more difficult to quantify and has challenged every one of us who purchases an investment, conceives a child, or flies an airplane – anyone who takes a risk.
Daniel Bernoulli advanced this concept by drawing a distinction between expected value and expected utility where the difference lay in the circumstances of the individual taking the risk or making the bet. A given value has a different utility depending on its effect on one’s life. Although the probabilities are the same for everyone, “the utility … is dependent on the particular circumstances of the person making the estimate …There is no reason to assume that … the risks anticipated by each …must be deemed equal in value.” A wager with a fixed reward, some sum of money, for instance, will have a fixed value but will have a different utility depending on the financial circumstances of the recipient. A pittance to a wealthy man may change the life of a poor man. A drink of water means nothing to the comfortable and survival to the parched. Bernoulli understood that, while the odds, the probability, what mathematicians call “expected value,” is the same for everyone, some of us will find greater utility in a positive outcome that is problematically equal for all of us and less utility on an unsuccessful outcome that is, similarly, equal in probability for all of us. The adventurous, whether an entrepreneur in search of a big score or a pilot unafraid of the darkness, places greater utility on a successful outcome while the more timid – or more cautious – among us find less utility in success compared to our fear of failure where we place the greater utility. Daniel Bernoulli understood that probability – value – is quantifiable mathematics, while utility is psychology and unique to the bettor. The expected value of the roll of two dice is always 7. The utility of the outcome depends on the circumstances, temperament, and desperation or daring of the one rolling the dice.
Here we have one of the fallacies of the way we teach risk to pilots. Tools such as personal minimums deny the simple fact that because of our personal circumstances some of us – and all of us at certain times – are willing to assume greater risk. Why? Because the utility of the outcome will at times be greater or lesser, shifting the risk vs. reward equation. Think of the Medivac pilot who, by assuming a slightly greater probability of a gruesome outcome – a crash – may substantially increase the likelihood of saving one or more lives. He may be justified in taking a flight in conditions that would be foolhardy on a simple pleasure flight. (Or, as an old friend and F4 combat pilot used to say to me, “George, there are some things we just don’t need to do in peacetime.”) The problem is that risk is not necessarily linear and we may fail to appreciate that both its probability and its consequences may be increasing more rapidly than we appreciate. And our ability to calculate the probability and appreciate the consequence of the outcome can be warped by aspects of our own personal psychology that we may not fully understand. I know it is heresy, but the fact is that personal minimums are a sliding scale. Unfortunately, it can be an exponential scale. As we assume more risk, the consequence can increase exponentially
Daniel Bernoulli, although one of the greatest mathematicians of his era, was less interested in the objective probability of an outcome than in the decision making process by which we evaluate risk. His concept of utility provided the support structure for the law of Supply and Demand, The Victorian era understanding of how markets work. Bernoulli’s theory of “expected utility” which has become a concept fundamental to economics even to this day is an attempt to quantify how much we desire or want an item or an outcome – and a recognition that an outcome may mean more to one than to another and more on one day than on another. Daniel Bernoulli gave us the concept of utility to describe the variability of an outcome’s benefit to a given bettor. Economists attempt to quantify expected value by the price which someone is willing to pay for the realization of his desire. Utility correlates with our desire, want, or need. Since desire cannot be measured directly, it is measured by what we will barter for it.
In economics that is the price we are willing to pay for the fulfillment of that desire. Think of that night you really, really wanted to fly home despite dangerous weather that, assessed without the degree to which you wanted to fulfill your desire to get home, would have been an easy no-go decision. That is the “expected utility” that can be the multiplier to the objective measurement of probability – and can leave us scattered across a mountain slope. This outcome too often proves to be the price we were willing to pay for the utility we sought.
Interestingly, I have observed over the years that when we pilots have access to better equipment, onboard Nexrad radar, for instance, we tend to use it not to increase our safety margin, but to increase our utility – to fly in weather we would have otherwise eschewed or to fly closer to weather we would have otherwise given a wide berth. We seem to find a level of risk with which we are comfortable and given better tools we stand closer to the fire because we feel that a clearer picture of its boundary gives us more warmth without an increase in a level of risk we have long since accepted. We choose to expand our utility by accepting no improvement in the probability. And it works most of the time. The problem arises when we kid ourselves that we are increasing our safety when we are in fact accepting no improvement in safety in pursuit of greater utility. It is often a zero sum game. More of one, less of the other. More utility, less safety. More safety, less utility. The choice is ours.
In recent decades, mathematicians, logicians, psychologists, philosophers, military strategists, investors, actuaries, and particularly economists, have broadened and sharpened our understanding of risk. Psychologist Daniel Kahneman earned a Nobel Prize in Economics for demonstrating that markets are driven by the often irrational choices of their participants, further complicating the arithmetic. Many of his theories are evident in the decisions we pilots sometimes make. We distort the true value of probability. Probability is traditionally calculated on a scale from 0 to 1, wherein 0 represents never and 1 represents always. We overweight lower probability, considering a probability of .01 to be much greater than 0, concerning ourselves too greatly about an unlikely outcome. We underweight higher probabilities, considering a probability of .99 to be much less than 1, or at least less than the math would support, and therefore concerning ourselves too little about a very likely outcome. This, particularly when the outcome is a loss rather than a reward, leads to self-deceptive behavior. If you have ever been frightened away from a flight by the mere mention of icing you may have overweighted a low probability. If you have ever talked yourself into a flight by concocting a complicated route penetrating a line of frontal thunderstorms or seizing upon and taking to heart a weather briefer’s few encouraging words in the midst of a cautionary report, you may have been underweighting a high probability of a serious outcome. Who among us has not cherry-picked the words we want to hear from a forecast? In either case we are engaging in this probability distortion. Most of us make decisions in ways we don’t fully understand, relying on heuristics and intuition.
The risk matrix we know today is based on mathematics derived and consequences understood by the mid-eighteenth century. Risk is the product of two factors: probability and outcome. Taken together, these provide the risk matrix that is the basis for any understanding of risk and reward – or, in our case as pilots, the obverse of risk and calamity.
We need the statistical data and the mathematics of probability, from both of which we can determine the likelihood of an event, good or bad, and, finally we need to understand the consequences of the various possible outcomes, not just in universal terms but as those consequences impact our own lives given our specific circumstances. Those elements of knowledge have evolved over thousands of years – and yet even today we misunderstand them all too often. We see it written in the NTSB reports.
Notes:
Many of the ideas herein were gleaned from or inspired by, Against the Gods The Remarkable Story of Risk, by Peter L. Bernstein, one of those rare books that marshals such a depth and breadth of knowledge that I find it difficult to conceive how one man could assimilate and contain it all and express it so gracefully.
Bernoulli, Daniel, 1738, “Specimen Theoriae Novae de Mensura Sortis (Exposition of a New Theory on the Measurement of Risk).” Translated from the Latin by Louise Sommer in Econometrica, Vol. 22, 1954, pp. 23-26.
Bernoulli, Jacob, 1713. Ars Conjectandi. Abstracted in Newman, 1988, pp. 1425-1432
In the words of my friend, fellow aviator and instructor and airline pilot, Ronney Moss, we each should fly one airplane at a time, preferably the one we are in. The same could be said of the one life we are given to live. To that end, I give you my Risk Matrix for Life. Apply it where you will.
| Event | Trivial | Problematic | Catastrophic |
| Likely | Give it a little thought | Buy insurance | Are you crazy? |
| Might happen | Relax | So, it’s a bad day | Don’t let it happen |
| Unlikely | Who cares? | Fugetaboutit | We all die sometime |
| But it feels really good | Go for it. | Worry about it tomorrow | You only live once |
