Archive for category Aerospace
The use of weighted-sum value matrices is a core component of many system-procurement and organizational decisions including risk assessments. In recent years the USAF has eliminated weighted-sum evaluations from most procurement decisions. They’ve done this on the basis that system requirements should set accurate performance levels that, once met, reduce procurement decisions to simple competition on price. This probably oversimplifies things. For example, the acquisition cost for an aircraft system might be easy to establish. But life cycle cost of systems that includes wear-out or limited-fatigue-life components requires forecasting and engineering judgments. In other areas of systems engineering, such as trade studies, maintenance planning, spares allocation, and especially risk analysis, multi-attribute or multi-criterion decisions are common.
Weighted-sum criterion matrices (and their relatives, e.g., weighted-product, AHP, etc.) are often criticized in engineering decision analysis for some valid reasons. These include non-independence of criteria, difficulties in normalizing and converting measurements and expert opinions into scores, and logical/philosophical concerns about decomposing subjective decisions into constituents.
Years ago, a team of systems engineers and I, while working through the issues of using weighted-sum matrices to select subcontractors for aircraft systems, experimented with comparing the problems we encountered in vendor selection to the unrelated multi-attribute decision process of mate selection. We met the same issues in attempting to create criteria, weight those criteria, and establish criteria scores in both decision processes, despite the fact that one process seems highly technical, the other one completely non-technical. This exercise emphasized the degree to which aircraft system vendor selection involves subjective decisions. It also revealed that despite the weaknesses of using weighted sums to make decisions, the process of identifying, weighting, and scoring the criteria for a decision greatly enhanced the engineers’ ability to give an expert opinion. But this final expert opinion was often at odds with that derived from weighted-sum scoring, even after attempts to adjust the weightings of the criteria.
Weighted-sum and related numerical approaches to decision-making interest me because I encounter them in my work with clients. They are central to most risk-analysis methodologies, and, therefore, central to risk management. The topic is inherently multidisciplinary, since it entails engineering, psychology, economics, and, in cases where weighted sums derive from multiple participants, social psychology.
This post is an introduction-after-the-fact, to my previous post, How to Pick a Spouse. I’m writing this brief prequel to address the fact that blog excerpting tools tend to use only the first few lines of a post, and on that basis, my post appeared to be on mate selection rather than decision analysis, it’s main point.
If you’re interested in multi-attribute decision-making in the engineering of systems, please continue now to How to Pick a Spouse.
Katz’s Law: Humans will act rationally when all other possibilities have been exhausted.
Bekhap’s Law asserts that brains times beauty equals a constant. Can this be true? Are intellect and beauty quantifiable? Is beauty a property of the subject of investigation, or a quality of the mind of the beholder? Are any other relevant variables (attributes) intimately tied to brains or beauty? Assuming brains and beauty both are desirable, Backhap’s Law implies an optimization exercise – picking a point on the reciprocal function representing the best compromise between brains and beauty. Presumably, this point differs for all evaluators. It raises questions about the marginal utility of brains and beauty. Is it possible that too much brain or too much beauty could be a liability? (Engineers would call this an edge-case check of Beckhap’s validity.) Is Beckhap’s Law of any use without a cost axis? Other axes? In practice, if taken seriously, Backhap’s Law might be merely one constraint in a multi-attribute decision process for selecting a spouse. It also sheds light on the problems of Air Force procurement of the components of a weapons system and a lot of other decisions. I’ll explain why.
I’ll start with an overview of how the Air Force oversees contract awards for aircraft subsystems – at least how it worked through most of USAF history, before recent changes in procurement methods. Historically, after awarding a contract to an aircraft maker, the aircraft maker’s engineers wrote specs for its systems. Vendors bid on the systems by creating designs described in proposals submitted for competition. The engineers who wrote the specs also created a list of a few dozen criteria, with weightings for each, on which they graded the vendors’ proposals. The USAF approved this criteria list and their weightings before vendors submitted their proposals to ensure the fairness deserved by taxpayers. Pricing and life-cycle cost were similarly scored by the aircraft maker. The bidder with the best total score got the contract.
A while back I headed a team of four engineers, all single men, designing and spec’ing out systems for a military jet. It took most of a year to write these specs. Six months later we received proposals hundreds of pages long. We graded the proposals according to our pre-determined list of criteria. After computing the weighted sums (sums of score times weight for each criteria) I asked the engineers if the results agreed with their subjective judgments. That is, did the scores agree with the subjective judgment of best bidder made by these engineers independent of the scoring process. Only about half of them were. I asked the team why they thought the score results differed from their subjective judgments.
They proposed several theories. A systems engineer, viewing the system from the perspective of its interactions and interfaces with the entire aircraft may not be familiar with all the internal details of the system while writings specs. You learn a lot of these details by reading the vendors’ proposals. So you’re better suited to create the criteria list after reading proposals. But the criteria and their weightings are fixed at that point because of the fairness concern. Anonymized proposals might preserve fairness and allow better criteria lists, one engineer offered.
But there was more to the disconnect between their subjective judgments of “best candidate” and the computed results. Someone immediately cited the problem of normalization. Converting weight in pounds, for example, to a dimensionless score (e.g., a grade of 0 to 100) was problematic. If minimum product weight is the goal, how you do you convert three vendors’ product weights into grades on the 100 scale. Giving the lowest weight 100 points and subtracting the percentage weight delta of the others feels arbitrary – because it is. Doing so compresses the scores excessively – making you want to assign a higher weighting to product-weight to compensate for the clustering of the product-weight scores. Since you’re not allowed to do that, you invent some other ad hoc means of increasing the difference between scores. In other words, you work around the weighted-sum concept to try to comply with the spirit of the rules without actually breaking the rules. But you still end up with a method in which you’re not terribly confident.
A bright young engineer named Hui then hit on a major problem of the weighted-sum scoring approach. He offered that the criteria in our lists were not truly independent; they interacted with each other. Further, he noted, it would be impossible to create a list of criteria that were truly independent. Nature, physics and engineering design just don’t work like that. On that thought, another engineer said that even if the criteria represented truly independent attributes of the vendors’ proposed systems, they might not be independent in a mental model of quality judgment. For example, there may be a logical quality composed of a nonlinear relationship between reliability, spares cost, support equipment, and maintainability. Engineering meets philosophy.
We spent lunch critiquing and philosophizing about multi-attribute decision-making. Where else is this relevant, I asked. Hui said, “Hmmm, everywhere?” “Dating!” said Eric. “Dating, or marriage?”, I asked. They agreed that while their immediate dating interests might suggest otherwise, all four were in fact interested in finding a spouse at some point. I suggested we test multi-attribute decision matrices on this particular decision. They accepted the challenge. Each agreed to make a list of past and potential future candidates to wed, without regard for the likelihood of any mutual interest the candidate might have. Each also would independently prepare a list of criteria on which they would rate the candidates. To clarify, each engineer would develop their own criteria, weightings, and scores for their own candidates only. No multi-party (participatory) decisions were involved; these involve other complex issues beyond our scope here (e.g., differing degrees of over/under-confidence in participants, doctrinal paradox, etc.). Sharing the list would be optional.
Nevertheless, on completing their criteria lists, everyone was happy to share criteria and weightings. There were quite a few non-independent attributes related to appearance, grooming and dress, even within a single engineer’s list. Likewise with intelligence. Then there was sense of humor, quirkiness, religious compatibility, moral virtues, education, type A/B personality, all the characteristics of Myers-Briggs, Eysenck, MMPI, and assorted personality tests. Each engineer rated a handful of candidates and calculated the weighted sum for each.
I asked everyone if their winning candidate matched their subjective judgment of who the winner should have been. A resounding no, across the board.
Some adherents of rigid multi-attribute decision processes address such disconnects between intuition and weighted-sum decision scores by suggesting that in this case we merely adjust the weightings. For example, MindTools suggests:
“If your intuition tells you that the top scoring option isn’t the best one, then reflect on the scores and weightings that you’ve applied. This may be a sign that certain factors are more important to you than you initially thought.”
To some, this sounds like an admission that subjective judgment is more reliable than the results of the numerical exercise. Regardless, no amount of adjusting scores and weights left the engineers confident that the method worked. No adjustment to the weight coefficients seemed to properly express tradeoffs between some of the attributes. I.e., no tweaking of the system ordered the candidates (from high to low) in a way that made sense to each evaluator. This meant the redesigned formula still wasn’t trustworthy. Again, the matter of complex interactions of non-independent criteria came up. The relative importance of attributes seems to change as one contemplates different aspects of a thing. A philosopher’s perspective would be that normative statements cannot be made descriptive by decomposition. Analytic methods don’t answer normative questions.
Interestingly, all the engineers felt that listing criteria and scoring them helped them make better judgments about the ideal spouse, but not the judgments resulting directly from the weighted-sum analysis.
Fact is, picking which supplier should get the contract and picking the best spouse candidate are normative, subjective decisions. No amount of dividing a subjective decision into components makes it objective. Nor does any amount of ranking or scoring. A quantified opinion is still an opinion. This doesn’t mean we shouldn’t use decision matrices or quantify our sentiments, but it does mean we should not hide behind such quantifications.
From the perspective of psychology, decomposing the decision into parts seems to make sense. Expert opinion is known to be sometimes marvelous, sometimes terribly flawed. Daniel Kahneman writes extensively on associative coherence, finding that our natural, untrained tendency is to reach conclusions first, and justify them second. Kahneman and Gary Klein looked in detail at expert opinions in “Conditions for Intuitive Expertise: a Failure to Disagree” (American Psychologist, 2009). They found that short-answer expert opinion can be very poor. But they found that the subjective judgments of experts forced to examine details and contemplate alternatives – particularly when they have sufficient experience to close the intuition feedback loop – are greatly improved.
Their findings seem to support the aircraft engineers’ views of the weight-sum analysis process. Despite the risk of confusing reasons with causes, enumerating the evaluation criteria and formally assessing them aids the subjective decision process. Doing so left them more confident about their decisions, for spouse and for aircraft system, though those decision differed from the ones produced by weighted sums. In the case of the aircraft systems, the engineers had to live with the results of the weighted-sum scoring.
I was one of the engineers who disagreed with the results of the aircraft system decisions. The weighted-sum process awarded a very large contract to the firm whose design I judged inferior. Ten years later, service problems were severe enough that the Air Force agreed to switch to the vendor I had subjectively judged best. As for the engineer-spouse decisions, those of my old engineering team are all successful so far. It may not be a coincidence that the divorce rates of engineers are among the lowest of all professions.
Hedy Lamarr was granted a patent for spread-spectrum communication technology, paving the way for modern wireless networking.
Years ago in a meeting on design of a complex, redundant system for a commercial jet, I referred to probabilities of various component failures. In front of this group of seasoned engineers, a highly respected, senior member of the team interjected, “I don’t believe in probability.” His proclamation stopped me cold. My first thought was what kind a backward brute would say something like that, especially in the context of aircraft design. But Willie was no brute. In fact he is a legend in electro-hydro-mechanical system design circles; and he deserves that status. For decades, millions of fearless fliers have touched down on the runway, unaware that Willie’s expertise played a large part in their safe arrival. So what can we make of Willie’s stated disbelief in probability?
Friends and I have been discussing risk science a lot lately – diverse aspects of it including the Challenger disaster, pharmaceutical manufacture in China, and black swans in financial markets. I want to write a few posts on risk science, as a personal log, and for whomever else might be interested. Risk science relies on several different understandings of risk, which in turn rely on the concept of probability. So before getting to risk, I’m going to jot down some thoughts on probability. These thoughts involve no computation or equations, but they do shed some light on Willie’s mindset. First a bit of background.
Oddly, the meaning of the word probability involves philosophy much more than it does math, so Willie’s use of belief might be justified. People mean very different things when they say probability. The chance of rolling a 7 is conceptually very different from the chance of an earthquake in Missouri this year. Probability is hard to define accurately. A look at its history shows why.
Mathematical theories of probability only first appeared in the late 17th century. This is puzzling, since gambling had existed for thousands of years. Gambling was enough of a problem in the ancient world that the Egyptian pharaohs, Roman emperors and Achaemenid satraps outlawed it. Such legislation had little effect on the urge to deal the cards or roll the dice. Enforcement was sporadic and halfhearted. Yet gamblers failed to develop probability theories. Historian Ian Hacking (The Emergence of Probability) observes, “Someone with only the most modest knowledge of probability mathematics could have won himself the whole of Gaul in a week.”
Why so much interest with so little understanding? In European and middle eastern history, it seems that neither Platonism (determinism derived from ideal forms) nor the Judeo/Christian/Islamic traditions (determinism through God’s will) had much sympathy for knowledge of chance. Chance was something to which knowledge could not apply. Chance meant uncertainty, and uncertainty was the absence of knowledge. Knowledge of chance didn’t seem to make sense. Plus, chance was the tool of immoral and dishonest gamblers.
The term probability is tied to the modern understanding of evidence. In medieval times, and well into the renaissance, probability literally referred to the level of authority – typically tied to the nobility – of a witness in a court case. A probable opinion was one given by a reputable witness. So a testimony could be highly probable but very incorrect, even false.
Through empiricism, central to the scientific method, the notion of diagnosis (inference of a condition from key indicators) emerged in the 17th century. Diagnosis allowed nature to be the reputable authority, rather than a person of status. For example, the symptom of skin spots could testify, with various degrees of probability, that measles had caused it. This goes back to the notion of induction and inference from the best explanation of evidence, which I discussed in past posts. Pascal, Fermat and Huygens brought probability into the respectable world of science.
But outside of science, probability and statistics still remained second class citizens right up to the 20th century. You used these tools when you didn’t have an exact set of accurate facts. Recognition of the predictive value of probability and statistics finally emerged when governments realized that death records had uses beyond preserving history, and when insurance companies figured out how to price premiums competitively.
Also around the turn of the 20th century, it became clear that in many realms – thermodynamics and quantum mechanics for example – probability would take center stage against determinism. Scientists began to see that some – perhaps most – aspects of reality were fundamentally probabilistic in nature, not deterministic. This was a tough pill for many to swallow, even Albert Einstein. Einstein famously argued with Niels Bohr, saying, “God does not play dice.” Einstein believed that some hidden variable would eventually emerge to explain why one of two identical atoms would decay while the other did not. A century later, Bohr is still winning that argument.
What we mean when we say probability today may seem uncontroversial – until you stake lives on it. Then it gets weird, and definitions become important. Defining probability is a wickedly contentious matter, because wildly conflicting conceptions of probability exist. They can be roughly divided into the objective and subjective interpretations. In the next post I’ll focus on the frequentist interpretation, which is objective, and the subjectivist interpretations as a group. I’ll look at the impact of accepting – or believing in – each of these on the design of things like airliners and space shuttles from the perspectives of Willie, Richard Feynman, and NASA. Then I’ll defend my own views on when and where to hold various beliefs about probability.
On reading my praise of Richard Feynman, a fellow systems engineer and INCOSE member (International Council on Systems Engineering) suggested that I read Feynman’s Minority Report to the Space Shuttle Challenger Enquiry. He said I might not like it. I read it, and I don’t like it, not from the perspective of a systems engineer.
Challenger explosion, Jan. 28, 1986
I should be clear on what I mean by systems engineering. I know of three uses of the term: first, the engineering of embedded systems, i.e., firmware (not relevant here); second, an organizational management approach (relevant, but secondary); third, a discipline aimed at design of assemblies of components to achieve a function that is greater than those of its constituents (bingo). Definitions given by others are useful toward examining Feynman’s minority report on the Challenger.
Simon Ramo, the “R” in TRW and inventor of the ICBM, put it like this: “Systems engineering is a discipline that concentrates on the design and application of the whole (system) as distinct from the parts. It involves looking at a problem in its entirety, taking into account all the facets and all the variables and relating the social to the technical aspect.”
Howard Eisner of GWU says, “Systems engineering is an iterative process of top-down synthesis, development, and operation of a real-world system that satisfies, in a near optimal manner, the full range of requirements for the system.”
INCOSE’s definition is pragmatic (pleasantly, as their guide tends a bit toward strategic-management jargon): “Systems engineering is an interdisciplinary approach and means to enable the realization of successful systems.”
Feynman reaches several sound conclusions about root causes of the flight 51-L Challenger disaster. He observes that NASA’s safety culture had critical flaws and that its management seemed to indulge in fantasy, ignoring the conclusions, advice and warnings of diligent systems and component engineers. He gives specific examples of how NASA management grossly exaggerated the reliability of many systems and components in the shuttle. On this point he concludes, “reality must take precedence over public relations, for nature cannot be fooled.” He describes a belief by management that because an anomaly was without consequence in a previous mission, it is therefore safe. Most importantly, he cites the erroneous use of the concept of factor of safety around the O-ring seals between the two lower segments of the solid rocket motors by NASA management (the Rogers Commission also agrees that failure of these O-rings was the root cause of the disaster). An NASA report on seal erosion in an earlier mission (flight 51-C) had assigned a safety factor of three, based on the seals having eroded only one third of the amount thought to be critical. Feynman replies that the O-rings were not designed to erode, and hence the factor-of-safety concept did not apply. Seal erosion was a failure of the design, catastrophic or not; there was no safety factor at all. “Erosion was a clue that something was wrong; not something from which safety could be inferred.”
But later Feynman incorrectly states that establishing a hypothetical propulsion system failure rate of 1 in 100,000 missions would require an inordinate number of tests to determine with confidence. Here he seems not to grasp both the exponential impact of redundancy on reliability, and that fault tree analysis could confidently calculate low system failure rates based on historical failure rates of large populations of constituent components, combined with the output of FMEAs (failure mode effects analyses) on those components in the relevant systems. This error does not impact Feynman’s conclusions about the root cause of the Challenger disaster. I mention it here because Feynman might be viewed as an authoritative source on systems engineering, but is here doing a poor job of systems engineering.
Discussing the liquid fuel engines, Feynman then introduces the concept of top-down design, which he criticizes. It isn’t clear exactly what he means by top-down. The most charitable reading would be a critique of NASA top management’s overruling the judgments of engineering management and engineers; but, on closer reading, it’s clear this cannot be his meaning:
The usual way that such engines are designed (for military or civilian aircraft) may be called the component system, or bottom-up design. First it is necessary to thoroughly understand the properties and limitations of the materials to be used (for turbine blades, for example), and tests are begun in experimental rigs to determine those. With this knowledge larger component parts (such as bearings) are designed and tested individually…
The Space Shuttle Main Engine was handled in a different manner, top down, we might say. The engine was designed and put together all at once with relatively little detailed preliminary study of the material and components. Then when troubles are found in the bearings, turbine blades, coolant pipes, etc., it is more expensive and difficult to discover the causes and make changes.
All mechanical-system design is necessarily top-down, in the sense of top-down used by Eisner, above. This use of the term is metaphor for progressive functional decomposition from mission requirements down to component requirements. Engineers cannot, for example, size a shuttle’s fuel pumps based on the functional requirement of having five men and two women orbit the earth to deploy a communications satellite. The fuel pump’s performance requirements ultimately emerge from successive derivations of requirements for subsystem design candidates. This design process is top-down, whether the various layers of subsystem design candidates are themselves newly conceived systems or ones that are already mature products (“off the shelf”). Wikipedia’s article and several software methodology sites incorrectly refer to design using off-the-shelf components as bottom-up – not involving functional decomposition. They err by failing to consider that piecing together existing subsystems toward a grander purpose still first requires functional decomposition of that grander purpose into lower-level requirements that serve as a basis for selecting existing subsystems. Simply put, you’ve got to know what you want a thing to do, even if you build that thing from available parts – software or hardware – in order to select those parts. Using off-the-shelf software subsystems still requires functional decomposition of the desired grander system.
F-117 frontal view
Off-the-shelf is a common strategy in aerospace, primarily for cost and schedule reasons. The Lockheed F-117, despite its unique design, used avionics taken from the C-130 and the F-16, brakes from the F-15, landing gear from the T-38, and other parts from commercial and military aircraft. This was for expediency. For the F-117, these off-the-shelf components still had to go through the necessary requirements validation, functional and stress testing, certification, and approval by all of the “ilities” (reliability, maintainability, supportability, durability, etc) required to justify their use in the vehicle – just as if they were newly designed. Likewise for the Challenger, the choice of new design vs. off-the-shelf should have had no impact on safety or reliability if proper systems engineering occurred. Whether its constituents were new designs or off-the-shelf, the shuttle’s propulsion system is necessarily – and desirably – the result of top-down design. Feynman may simply mean that the design and testing phases were rushed, that omissions were made, and that testing was incomplete. Other evidence suggests this; but these omissions are not a negative consequence of top-down design, which is the only sound process for the design of aircraft and other systems of systems.
It is difficult to imagine any sound basis for Feynman’s use of – and defense of – bottom-up design other than the selection of off-the-shelf components, which, as mentioned above, still entails functional decomposition (top-down design). Other uses of the term appear in discussions of software methodologies. I also found a handful of academic papers that incorrectly – incoherently, in my view – equate top-down with analysis and deduction, and bottom-up with synthesis and induction. The erroneous equation of analysis with deductive reasoning pops up in Design Thinking and social science literature (e.g., at socialresearchmethods.net). It fails to realize that analysis as a means of inferring cause from observed result (i.e., what made this happen?) always entails inductive reasoning. Geometry is deduction; science and engineering are inherently inductive.
The use of bottom-up shows up in software circles in a disparaging sense. It describes a state of system growth that happens with no conscious design beyond that of an original seed. It is non-design, in a sense. Such “organic growth” happens in enterprise software when new features, not envisioned during the original design, are later bolted-on. This can stem from naïve mismanagement by those unaware of the damage done to maintainability and further extensibility of the software system, or through necessity in a merger/acquisition scenario where the system’s owners are aware of the consequences but have no other alternatives. This scenario obviously does not apply to the hardware or software of the Challenger; and if it did, such bottom-up “design” would be a defect of the system, not a virtue.
Hydro-mechanical system components in 737 gear bay
Aerospace has in its legacy an attitude – as opposed to a design method – sometimes called a bottom-up mindset. I’ve encountered this as a form of resistance to methodological system-design-for-safety and the application of redundancy. In my experience it came from expert designers of electro-hydro-mechanical subsystems. A legendary aerospace systems designer once told me with a straight face, “I don’t believe in probability.” You can trace this type of thinking back to the rough and ready pioneers of manned flight. Charles Lindbergh, for example, said something along the lines of, “give me one good engine and one good pilot.” Implicit in this mentality is the notion that safety emerges from component quality rather than from system design. The failure rates of the best aerospace components tend to vary from those of average components by factors of two or ten, whereas redundancy has an exponential effect. Feynman’s criticism of top-down and endorsement of bottom-up – whatever he meant by it – could unfortunately be seen as support for this harmful and oddly persistent notion of bottom-up.
Toward the end of Feynman’s report, he reveals another misunderstanding about design of life-critical systems. In the section on avionics, he faults NASA for using 15-year-old software and hardware designs, concluding that the electronics are obsolete. He claims that modern chip sets are more reliable and of higher quality. This criticism runs contrary to his complaint about top-down design of the main engines, and it misses a key point. The improvements in reliability of newer chips would contribute only negligibly toward improved availability of the quad-redundant system containing them. More importantly, older designs of electronic components are often used in avionics precisely because they are old, mature designs. Accelerated-life testing of electronics is known to be tricky business. We use old-design chips because there is enough historical usage data to determine their failure rates without relying on accelerated-life testing. Long ago at McDonnell Douglas I oversaw use of the Intel 87C196 chip for a system on the C-17 aircraft. The Intel rep told me that this was the first use of the Intel 8086-derivative chip in a military aircraft. We defended its use, over the traditional but less capable Motorola chips, on the basis that the then 10+ year history of 8086’s in similar environments was finally sufficient to establish a statistical failure rate usable in our system availability calculations. Interestingly, at that time NASA had already been using 8086 chips in the shuttle for years.
Feynman’s minority report on the Challenger contains misunderstandings and technical errors from the perspective of a systems engineer. While these errors may have little impact on his findings, they should be called out because of the possible influence they may have on future generations of engineers. The tyranny of pedigree, as we saw with Galileo, can extend a wrong idea’s life for generations.
That said, Feynman makes several key points about the psychology of engineering management that deserve much more attention than they get in engineering circles. First among these in my mind is the fallacy of induction from near-misses viewed as successes, thereby producing undue confidence about future missions.
“His legs were weary, but his mind was at ease, free from the presentiment of change. The sense of security more frequently springs from habit than from conviction, and for this reason it often subsists after such a change in the conditions as might have been expected to suggest alarm. The lapse of time during which a given event has not happened is, in the logic of habit, constantly alleged as a reason why the event should never happen, even when the lapse of time is precisely the added condition which makes the event imminent. A man will tell you that he has worked in a mine for forty years unhurt by an accident, as a reason why he should apprehend no danger, though the roof is beginning to sink; and it is often observable that the older a man gets, the more difficult it is to retain a believing conception of his own death.”
– from Silas Marner, by George Eliot (Mary Ann Evans Cross), 1861
Text and aircraft photos copyright 2013 by William Storage. NASA shuttle photos public domain.
In a recent post I wrote about Vic, who might not look like a Christian, but probably is one. The Vic example reminded me of a famous study of unintuitive probabilities done in 1983. Amos Tversky and Daniel Kahneman surveyed students at the University of British Columbia using something similar to my Vic puzzle:
Linda is 31 years old, single, outspoken, and very bright. She majored in philosophy. As a student, she was deeply concerned with issues of discrimination and social justice, and also participated in anti-nuclear demonstrations.
Which is more probable?
A. Linda is a bank teller.
B. Linda is a bank teller and is active in the feminist movement.
About 90% of students said (B) was more probable. Mathematicians point out that, without needing to know anything about Linda, (A) has to be more probable than (B). Thinking otherwise is the conjunction fallacy. It’s simple arithmetic. The probability of a conjunction, P(A&B), cannot exceed the probabilities of its constituents, P(A) and P(B), because the extension (possibility set) of the conjunction is included in the extension of its constituents. In a coin toss, the probability of heads has to exceed the probability of heads AND that it will rain today.
Putting numbers to Linda, one might guess there’s 1% probability that Linda, based on the description given, is a bank teller, but a 99% probability that she’s a feminist. Even so, 1% is still a bigger number (probability) than 1% AND 99%, which means 1% times 99% – which is a tad less than 1%.
So why does it seem like (B) is more likely? Lots of psychological and semantic reasons have been proposed. For example, in normal communications, we usually obey some unspoken principle of relevance; a sane person would not mention Linda’s marital status, political views and values if they were irrelevant to the question at hand – which somehow seems to have something to do with Linda’s profession. Further, humans learn pattern recognition and apply heuristics. It may be a fair bit of inductive reasoning based on past evidence that women active in the feminist movement are more likely than those who are not to major in philosophy, be single, and be concerned with discrimination. This may be a reasonable inference, or it may just prove you’re a sexist pig for even thinking such a thing. I attended a lecture at UC Berkeley where I was told that any statement by men that connects attributes (physical, ideological or otherwise) to any group (except white men) constituted sexism, racism or some otherism. This made me wonder how feminists are able to recognize other feminists.
In any case, there are reasons that student would not give the mathematically correct answer about Linda beyond the possibility that they are mathematically illiterate. Tversky and Kahneman tried various wordings of the problem, pretty much getting the same results. At some point they came up with this statement of the problem that seems to drive home the point that they were seeking a mathematical interpretation of the problem:
Argument 1: Linda is more likely to be a bank teller than she is to be a feminist bank teller, because every feminist bank teller is a bank teller, but some bank tellers are not feminists, and Linda could be one of them.
Argument 2: Linda is more likely to be a feminists bank teller than she is likely to be a bank teller, because she resembles an active feminist more than she resembles a bank teller.
In this case 65% of students chose the extension argument (2), despite its internal logical flaw. Note that argument 1 explains why the conjunction fallacy is invalid and that argument 2 doesn’t really make much sense.
Whatever the reason we tend to botch such probability challenges, there are cases in engineering that are surprisingly analogous to the Linda problem. For example, when building a fault tree (see fig. 1), your heuristics can make you miss event dependencies and common causes between related failures. For example, if an aircraft hydraulic brake system accumulator fails by exploding instead of by leaking, and in doing so severs a hydraulic line, an “AND” relationship disappears so that what appeared to be P(A&B) becomes simply P(A). Such logic errors can make calculations of probability of catastrophe off by factors of thousands or millions. This is bad, when lives are at stake. Fortunately, engineers apply great skill and discipline to modeling this sort of thing. We who fly owe our lives to good engineers. Linda probably does too.
Fig. 1. Segment of a fault tree for loss of braking in a hypothetical 8-wheeled aircraft using FTA software I authored in 1997. This fault tree addresses only a single Class IV hazard in aircraft braking – uncontrolled departure from the end of the runway due to loss of braking during a rejected takeoff. It calculates the probability of this “top event” as being more remote than the one-per-billion flight hours probability limit specified by the guidelines of FAA Advisory Circular 25.1309-1A, 14CFR/CS 25.1309, and SAE ARP4754. This fault tree, when simplified by standard techniques, results in about 200,000 unique cut sets – combinations of basic events leading to the catastrophic condition.
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Uncertainty is an unavoidable aspect of the human condition- Opening sentence of “Extensional Versus Intuitive Reasoning” by Tversky and Kahneman, Oct. 1983 Psychological Review.