Education – Mike Trick's Operations Research Blog

Helping Operations Researchers start young

Now that my son has turned twelve, I am beginning to see among his friends a bifurcation: some “love” mathematics and some “hate” mathematics. This is frustrating to me since I know that if the kids see operations research then they will all love it. What is not to like? Coloring maps, finding paths, scheduling sports leagues…. who could ask for more? But if they hate mathematics now, it will be very hard to get them into operations research in the future.

My son has generally had good mathematics teachers, and his curriculum is full of interesting aspects of mathematics. Sometimes the variety drives me crazy (seriously: shouldn’t a twelve year old know what 9×7 is without hemming and hawing?) but most of the time I like what he is learning. Rather than rote memorization, the teachers are trying to imbue him with an understanding of concepts such as measurement, algebraic thinking, counting, geometry and so on. I do think such an approach to mathematics will lead to a greater appreciation for things like operations research when the time comes. And I wish that time was no later than high school (see HSOR for some older but still very useful modules for that level).

I have a colleague in New Zealand who is trying to provide rich mathematical-oriented resources for a variety of levels. Nicola Petty (“Dr. Nic”) runs the Statistics Learning Center which has, among many other things, a number of very useful videos on many statistical concepts. I often send my students there when it appears my own explanations of those concepts is not meshing with them.

cat Dr. Nic has a Kickstarter going on for a set of mathematics cards aimed at the 5-8 year old set. Each “Cat” card describes a cat along with her possessions, age, characteristics and so on. Kids can use these cards to explore issues in counting, graphing, statistics, and many other areas. I haven’t found a way to naturally embed a shortest path problem in the set, but perhaps the more creative of you can create a game that illustrates the concept of NP-completeness (and, coincidentally, keeping the kids quiet for a very long time).

I would hope that items like this (along with all the other things Dr. Nic offers) would spark interest in “our” type of mathematics. If you have a kids of the appropriate age (or are a kid of that age who happens to read Operations Research blogs) or know someone or a school that could use this resource, I hope you can support her Kickstarter or her other offerings. And look for a card named “Trafalgar” (I hope) in the final edition of the Cat cards!

Taking Optimization With You After Graduation

In the Tepper MBA program, we use versions of Excel’s Solver (actually a souped up version from Frontline Systems) for most of our basic optimization courses. Students like this since they feel comfortable with the Excel interface and they know that they can use something like this in their summer internships and first jobs, albeit they are likely to the more crippled version standard with Excel. For those who are particularly keen, we point them to an open source optimization system that can allow them to stay within the Excel structure.

In our most advanced course, we use AIMMS with Gurobi as the underlying solver. Students generally love the power of the system, but worry that they will not be able to translate what they learn into their jobs. This wouldn’t be an issue if companies had analytics and optimization as a core strength, and routinely had some of the commercial software, but that is not the case. So the issue of transfer comes up often.

I am really happy to see that Gurobi has a deal in place to allow students to continue using their software, even after they graduate. This gives new graduates some time to wow their new employers with their skills, and to make the argument for further investment in operations research capabilities.

Here is an excerpt from an email I received from Gurobi:

Academic Site License

Our FREE academic site license allows students, faculty, and staff at a degree-granting academic institution to use Gurobi from any machine on a university local-area network. This program makes it easy for everyone at the university to have access to the latest version of Gurobi, without having to obtain their own license. You can learn more on our Academic Licensing page, and your network administrator can request a license by emailing support@gurobi.com.

Take Gurobi With You Program Update

This program allows qualified recent graduates to obtain a FREE one-year license of Gurobi for use at their new employer.

Qualified recent graduates can complete a short approval process and then receive the license including maintenance and support at no cost to themselves or their employers. This reflects our continuing support of students, even after they graduate. You can learn more on our Take Gurobi With You page.

I think this sort of program can have a great effect on the use of optimization in practice. And we need to rethink what we teach in the classrooms now that we know the “can’t take it with you” effect is lessened.

Blogging and the Changing World of Education

As a blogger, I have been a failure in the last six months. I barely have enough time to tweet, let alone sit down for these extensively researched, tightly edited, and deeply insightful missives that characterize my blog. I tell you, 1005 words on finding love through optimization doesn’t just happen!

I have my excuses, of course. As the fabulous PHD Comics points out, most of us academics seem somewhat overbooked, despite the freedom to set much of our schedule. I am not alone in being congenitally unable to turn down “opportunities” when they come by. “Help hire a Norwegian professor?” Sounds fun! “Be the external examiner for a French habilitation degree?” I am sure I’ll learn a lot! “Referee another paper?” How long can that take? “Fly to Australia for a few days to do a research center review?” Count me in! And that was just four weeks in February.

All this is in addition to my day job that includes a more-than-healthy dose of academic administration. Between doing my part to run a top business school and to move along in research, not to mention family time, including picking up the leavings of a hundred pound Bernese Mountain Dog (the “Mountain” in the name comes from said leavings) and entertaining a truly remarkable nine-year-old son, my time is pretty well booked up.

And then something new comes along. For me, this newness is something I had a hand in putting together: the Tepper School’s new FlexMBA program. This program offers our flagship MBA program in a hybrid online/onsite structure. Every seven weeks or so, students in the program gather at one of CMU’s campuses (we have them in Pittsburgh, Silicon Valley, and New York, we have not yet used our Qatar campus) and spend a couple days intensively starting their new courses. This is followed by six weeks of mixed synchronous and asynchronous course material. Asynchronous material is stuff the students can do in their own time: videos, readings, assignments, and so on. The synchronous lesson is a bit more than an hour in a group, meeting via a group video conference, going over any issues in the material and working on case studies, sample problems, and so on. The course ends with exams or other evaluations back on campus before starting the next courses.

Our commitment is to offer the same program as our full-time residential MBA and our part-time in-Pittsburgh MBA. So this means, the same courses, faculty, learning objectives, and evaluations that our local students take.

We started this program last September with 29 students, and so far it has gone great. The students are highly motivated, smart, hard-working, and engaged. And the faculty have been amazing: they have put in tons of work to adapt their courses to this new structure. Fortunately, we have some top-notch staff to keep things working. Unlike some other MBA programs, we have not partnered with any outside firm on this. If we are going to offer our degree, we want it to be our degree.

I have just finished my own course in this program. I teach our “Statistical Decision Making” course. This is a core course all MBA students take and revolves around multiple regression and simulation (the interesting relationships between these topics can wait for another day). This is not the most natural course for me: my research and background is more on the optimization side, but I very much enjoy the course. And teaching this course has made clear to me the real promise of the hot phrase “business analytics”: the best of business analytics will combine the predictive analytics of statistics and machine learning with the prescriptive analytics of optimization, again a topic for another day.

My initial meeting with the students concentrated on an overview of the course and an introduction to the software through some inspiring cases. We then moved on the the six-week distance phase. Each of the six modules that make up a course is composed of four to eight topics. For instance, one of my modules on multiple regression includes the topic “Identifying and Handling Muliticollinearity”. (Briefly: multicollearity occurs when you do regression with two or more variables that can substitute for each other; imagine predicting height using both left-foot-length and right-foot-length as data). That section of the module consists of

A reading from their textbook on the subject
One 8 minute video from me on “identifying multicollinearity”
One 6 minute video from me on “handling multicollinerity”
A three minute video of me using our statistical software to show how it occurs in the software (I separate this out so we can change software without redoing the entire course)
A question or two on the weekly assignment.

It would be better if I also had a quiz to check understanding of the topic, along with further pointers to additional readings.

So my course, which I previously thought of as 12 lectures, is now 35 or so topics, each with readings, videos, and software demonstrations. While there are some relationships between the topics, much is independent, so it would be possible, for instance, to pull out the simulation portion and replace it with other topics if desired. Or we can now repackage the material as some supplementary material for executive education courses. The possibilities are endless.

Putting all this together was a blast, and I now understand the structure of the course, how things fit together, and how to improve the course. For instance, there are topics that clearly don’t fit in this course, and would be better elsewhere in the curriculum. We can simply move those topics to other courses. And there are linkages between topics that I did not see before I broke down the course this finely.

I look forward to doing this for our more “operations research” type courses (as some of my colleagues have already done). Operations Research seems an ideal topic for this sort of structure. Due to its mathematical underpinnings and need for organized thinking, students sometimes find this subject difficult. By forcing the faculty to think about it in digestible pieces, I think we will end up doing a better job of educating students.

Creating this course was tremendously time consuming. I had not taken my own advise to get most of the course prepared before the start of the semester, so I was constantly struggling to stay ahead of the students. But next year should go easier: I can substitute out some of the videos, extend the current structure with some additional quizzes and the like, adapt to any new technologies we add to the program, and generally engage in the continuous improvement we want in all our courses.

But perhaps next year, I won’t have to take a hiatus from blogging to get my teaching done!

Operations Research at Business Schools: The Bad!

Right after returning from Egon Balas’ 90th Birthday tea, as I thought good thoughts about the role operations research plays in business schools, I read some disconcerting news from the College of of Business and Economics, University of Canterbury in Christchurch, New Zealand, from Dr. Nicola Petty:

On Wednesday the Council of the university at which I have been employed voted to close down the Operations Research programme. The university wants to “concentrate” and OR didn’t make the grade, despite two academics taking voluntary redundancy, and a concerted effort to streamline the programme so that it is financially viable. It is the end of an era.

Let me begin by saying that I love New Zealand and its universities. I spoke at a number of them in 2007 when I was the New Zealand Operational Research Society Visiting Speaker, and I spent a wonderful couple of days at the University of Canterbury. Christchurch is a beautiful town that has had a tough time of it recently. And I have no direct knowledge of the challenges the university faces, or what went into the decision.

That said, I have to ask:

What the heck is that university thinking?????

Here we are, entering a golden age of analytical decision making. We are in a world where companies are drowning in data but are unable to make sense of it to turn data into better decisions. Where companies like IBM put business analytics front and center in their strategic plans. Where a key managerial skill is understanding data and applying analytical approaches to problems.

What kind of management program would purposefully cut their business analytics capabilities in this world?

Stunning.

As an academic administrator in a business school, I guess I am happy to see our “competitors” making themselves weaker. More for us, I guess.

As someone in operations research, it is depressing to see how some academic administrators just don’t get it. It gives the rest of us academic administrators a bad name.

If you are a student planning to study management, please ask the question: am I going to get the skills I need to survive and thrive in a data-rich environment full of complicated decisions? A management school that is running away from analytics is a school that is living in the past.

Operations Research and Business Schools: The Good!

Way back in 1988, I was a fresh Ph.D. out of Georgia Tech doing a postdoc at the Institute for Mathematics and its Applications at the University of Minnesota. While I had plans to spend another postdoc year, likely in Europe (and I ended up doing so, in Germany), I did decide it would be good to lock up an academic job before I left. Email did exist at the time, but the norm was to send things out via “regular” mail. So I went down to the copy center at the University and picked out a suitably heavy-weight paper for my vita. I sent out a dozen or so responses to job ads and made a few phone calls (or asked my advisor to make a few calls) and was invited to visit a half-dozen or so places. Perhaps it was different era, or perhaps I was relaxed knowing that I had another year of postdoc funds if needed, but it certainly felt more relaxed that it appears to be these days.

One place that seemed eager to have me out was this “business school” at Carnegie Mellon: The Graduate School of Industrial Administration. Now, I came out of engineering and I certainly believed that my future lie in engineering. Here is a sign of how little those of us in engineering knew about business schools: the previous year, a fellow doctoral student went out on the market and interviewed at a number of places before finding a job at a business school. At the time, we were all a bit surprised since he had a good dissertation and we (other doctoral students) thought that it was good enough to get a top engineering job. Too bad he was stuck in a business school, we said: must be a tough job market. That school was the University of Chicago, then and now a preeminent business school that much of the field would kill to get a job at. Business schools were really not on our radar.

But I was polite, so I agreed to head out to Carnegie Mellon. It was my first job interview, so I told myself the school would be a great place to practice my talk before moving on to the real contenders.

I hadn’t planned on liking CMU and GSIA as much as I did. The people I talked to were much different than those at engineering schools. Of course, there were some top-notch OR people (more on them later) but I also talked to economists and political scientists and even a psychologist or two. They were involved in fascinating research that was a little less … transactional than much of engineering research (“Do this since the grant depends on it”). And the Deputy Dean of the time, Tim McGuire (now at Management Science Associates) was very persuasive about how exciting things can be in business schools.

But even more persuasive was Egon Balas, an intellectual leader in the operations research since the 1960s. While I did (and do) find him a bit intimidating, Egon had (and has) a tremendous love for integer programming, and amazing energy in research. He also had spend decades keeping up the tradition GSIA had of having a great OR group. Founders such as Herb Simon, Bill Cooper, Al Blumstein, and Gerry Thompson had been (or in Gerry’s case, still were) part of GSIA, and the OR group was, in 1988, pretty amazing: Gerard Cornuejols and John Hooker joined Gerry and Egon to form the group.

I received an offer from GSIA and from some top engineering schools, and, to my surprise, I decided that my future lay in the business school. And that is not a decision I have regretted. GSIA (now Tepper) continues to have a top-notch OR group. Gerry retired, then passed away, but we added R. Ravi, Javier Pena, Francois Margot, Willem van Hoeve, and Fatma Kilinc-Karzan. Gerard Cornuejols continues to do amazing work, having recently won the von Neumann Theory Prize.

With the larger faculty size comes a stable and important role within the business school. Operations research is seem as a key competitive advantage to our school. While there are many aspects of this advantage, I’ll point to two: the increased role of business analytics, and the role rankings play in business school success. If you don’t believe me on the latter, I’ll point you to the list of journals Business Week uses for their intellectual capital ranking. If you have people who can publish in Operations Research, you can be a more successful business school. I recently heard my Dean, a hard-core finance researcher, say “We need more OR faculty”: music to my ears!

And, the best part is, Egon Balas is still with us and still active. He turns 90 this week, so we had a tea for him (we had a big conference when he turned 80; we can do that again for his 100th). A bunch of us did short video clips to wish Egon happy birthday. Here is mine:

As you might guess, I am proud to be part of the operations research group here at the Tepper School. The school has been very good for operations research … and operations research has been very good for the school.

Teaching and Research

For the last few years, I have been dabbling in academic administration, first as Associate Dean for Research and now as Senior Associate Dean, Education here at the Tepper School of Business. While there are frustrations in this position (“There are how many courses not covered? And are all the adjuncts on vacation in Aruba now?”), some aspects are wonderful. Working with new faculty is a great pleasure, a pleasure that alone almost offsets the hassles. I love the excitement and the energy and the feeling that anything is possible.

This was easy on the research side of the organization: my job was to create a great research environment (subject to resource constraints, of course!), and that was very rewarding to do. On the education side, my job is a bit different. While some faculty love teaching, for others it seems to take time away from what they really want to do: research. How can they do any research if they have to do any teaching?

Teaching is hard, and takes time and energy. Does it take time away from research? While I can talk to new faculty about how teaching and research intersect, and how one builds on the other, I can see a fair amount of eye-rolling. Of course, I would say that: that’s my job! And when I explain that the entire “sports scheduling” part of my career happened due to an offhand conversation with an MBA student, the response is a mixture of “That’s what I have to look forward to? Sports Scheduling?” and “Sure, teaching might be OK for practical types, but what about us theory types?”

Thanks to a colleague (thanks Stan!), I think I now have the perfect riposte. This is from Richard Feynman‘s “Surely you’re joking, Mr. Feynman!”:

I don’t believe I can really do without teaching. The reason is, I have to have something so that when I don’t have any ideas and I’m not getting anywhere I can say to myself, “At least I’m living; at least I’m doing something; I am making some contribution” — it’s just psychological.

When I was at Princeton in the 1940s I could see what happened to those great minds at the Institute for Advanced Study, who had been specially selected for their tremendous brains and were now given this opportunity to sit in this lovely house by the woods there, with no classes to teach, with no obligations whatsoever. These poor bastards could now sit and think clearly all by themselves, OK? So they don’t get any ideas for a while: They have every opportunity to do something, and they are not getting any ideas. I believe that in a situation like this a kind of guilt or depression worms inside of you, and you begin to worry about not getting any ideas. And nothing happens. Still no ideas come.

Nothing happens because there’s not enough real activity and challenge: You’re not in contact with the experimental guys. You don’t have to think how to answer questions from the students. Nothing!

In any thinking process there are moments when everything is going good and you’ve got wonderful ideas. Teaching is an interruption, and so it’s the greatest pain in the neck in the world. And then there are the longer period of time when not much is coming to you. You’re not getting any ideas, and if you’re doing nothing at all, it drives you nuts! You can’t even say “I’m teaching my class.”

If you’re teaching a class, you can think about the elementary things that you know very well. These things are kind of fun and delightful. It doesn’t do any harm to think them over again. Is there a better way to present them? The elementary things are easy to think about; if you can’t think of a new thought, no harm done; what you thought about it before is good enough for the class. If you do think of something new, you’re rather pleased that you have a new way of looking at it.

The questions of the students are often the source of new research. They often ask profound questions that I’ve thought about at times and then given up on, so to speak, for a while. It wouldn’t do me any harm to think about them again and see if I can go any further now. The students may not be able to see the thing I want to answer, or the subtleties I want to think about, but they remind me of a problem by asking questions in the neighborhood of that problem. It’s not so easy to remind yourself of these things.

So I find that teaching and the students keep life going, and I would never accept any position in which somebody has invented a happy situation for me where I don’t have to teach. Never.

If not teaching ruined the minds of those at the Institute for Advanced Study, imagine the effect on us mere mortals! So teach, already! And if you want to teach a bit extra, I happen to have a few courses that need to be covered….

Puppetry, Turf Management, and Operations Research

CNN and careerbuilder.com have put out a list of six unusual college degrees. I checked it out, expecting to see Carnegie Mellon’s own offering in this area: bagpiping. But bagpiping was not unusual enough to make this list. After possibilities for racetrack management and packaging (“Don’t think outside the box: think about the box”), there was one appealing one nestled between puppetry and turfgrass management: “decision making”. At the Kelly School at the University of Indiana, you can get a doctorate in “help[ing] future business leaders analyze and make decisions.” Wow!

Of course, this is just our favorite field of Operations Research, weakly disguised with a fake mustache and beard:

According to the program’s website, “Decision Sciences is devoted to the study of quantitative methods used to aid decision making in business environments. Using mathematical models and analytical reasoning, students examine problems … and learn how to solve these problems by using a number of mathematical techniques, including optimization methods (linear, integer, nonlinear), computer simulation, decision analysis, artificial intelligence and more.”

In our never-ending quest to find the right name for our field, we are showing up on lists of wacky degrees, displacing bagpiping and cereal science (“Ingrain yourself to a great career”). Better that than being on no lists at all. Maybe a prospective puppeteer will see the list and decide to go into “decision making” instead. No strings attached.

Thanks to Kevin Furman for the pointer!

The List Every University Should Want to Be On

About this time last year, I asked advice about where a high school senior should consider for college engineering. At the back of my mind, I was figuring that this is a pretty smart kid and both of his parents have PhDs, so finding a place that might get him fired up enough to consider education beyond the bachelor’s degree would be a good idea. But what places really have an environment that inspires students to go on to still-higher education? While I had thought a smaller school would be a good choice, I also bought into the argument that students are inspired by top-notch research around them, arguing for a larger research-oriented institution.

The NSF has done a study on this, and is listing the undergraduate programs whose students are most likely to go on to get a PhD in a science or engineering field. Thanks to Laura McLay for pointing to this article on the subject. Here is the table that lists the top 50 schools in terms of fraction of students who go on to get a science and engineering doctorate:

The results are pretty stunning for a lot of reasons:

The list is dominated by private schools, with only three public schools in the top 50.
About half the schools are “Research – Very High”, denoting the most active research institutions (there are three levels of research activity in the Carnegie Classification), with the other half being small undergraduate-oriented colleges.
The effect is significant, with a factor of seven difference between number 1 and number 50 on the list. The average “Research – High” university has a value of 1.5 (1.5% go on to doctorates), so the lift for number 50 is more than 3, while that for Cal Tech is 23 (a graduate of Cal Tech is 23 times more likely to go on for a science and engineering doctorate than a graduate of an average research university).
Berkeley (who also has the largest number of graduates with PhDs) is the only large public university on the list. No Michigan (who has the second third largest number of graduates with PhDs) , Georgia Tech, or other large public university is to be seen.

Now this sort of study has limitations. In retrospect, it is not surprising that a school that graduates lots of, say, accountants will naturally have a lower fraction who go on to get doctorates. Most big public universities have honors programs or other structures to nurture those with further educational aspirations, and I am sure that the results for those in these specialized programs look like the results for the schools in the table above.

But if you want to be surrounded by those likely to go on to get doctorates in science and engineering, you should either go to one of the very top private research schools or go to a small private liberal arts college. Carnegie Mellon or Oberlin (or, particularly, Cal Tech), that is the question!

More Operations Research in the News, but not in a Welcome Way

Fabrice Tourre, “the fabulous Fab”, who is at the center of the Goldman Sachs scandal, is a 2001 graduate of Stanford University. That, in itself, is no surprise. Stanford has a top ranked business school that does about as well as the rest of us in graduating ethical MBAs (by that I mean MBAs who do, on the whole, try to act ethically, but some of whom find ethical challenges … challenging), so it is not surprising that a powerful Goldman Sachs person would come from there. But what is surprising is that Fabrice’s Stanford degree is a Masters in Operations Research! Our field is in the news!

Thinking about it, it is not so surprising. Since Fabrice is reported to be 31, a 2001 graduate would have been 22. Most business schools like to see at least a little work experience, so 22 year-olds with an MBA would be quite unusual. A Masters in Operations Research would be more common, I would think.

I can’t tell if the fabulous Fab did anything wrong, let alone illegal, but this does bring up an issue in training. At business schools, we are working hard to think about how to include ethics and other aspects of corporate social responsibility into the curriculum (with varying levels of success). What are operations research programs doing to ensure that their masters graduates are aware of the choices they make? Checking Georgia Tech, Michigan, Stanford Management Science and Engineering (is there still an MSOR from Stanford?), and Cornell (not to pick on them, but to pick a few of the best programs out there), does not lead one to believe that ethics, corporate responsibility or a traditional “engineering professional responsibility” course is part of the masters curriculum. This is not to suggest that we are putting out a generation of unethical lying optimizers, but perhaps we should rethink the balance of our programs. I do believe operations research to be outstanding training for a wide variety of careers: going beyond linear and integer programming into some of the challenges of the real world would be a good direction to go for the sake of the students, and for the rest of us.

Conditional Probability in the New York Times

When you ask a question of the form “What are the chances of X given Y”, your are asking a question of conditional probability. These sorts of questions have come up in this blog before: “What are the chances of cancer given a positive test result?” “What are the chances a monkey prefers blue M&Ms to green M&Ms, given it prefers red M&Ms to blue M&Ms?” “What are the chances of predicting the NCAA tournament perfectly, given perfect predictions for the first two rounds?”

Conditional probability is extremely important for two reasons. First, it occurs all the time: it is a fundamental building block as we aggregate information in an uncertain environment. Second, people are really, really bad at it. In case after case, our intuition misleads us and we badly misestimate conditional probabilities. When a 90% accurate drug test (meaning it is positive 90% of the time for a drug user, and negative 90% of the time for a nonuser) comes back positive, what is the probability the person uses drugs. Our intuition screams “It has to be 90%”! But the probability of “User given positive drug test” is not the same as probability of “positive drug test given user”. If 5% of the population use drugs, then the probability of “User given positive drug test is about 1 in 3. Consider 1000 people: 50 are drug users so 45 will test positive; of the 950 non-users, 95 will test positive; so 45/(45+95) is the probability of user given positive test.

Note that in the argument above, I did not rely on the main theorem in conditional probability: Bayes Theorem. Bayes Theorem states P(A|B) (the probability of A given B) = P(B|A)P(A)/P(B). I could have worked it out that way, but in doing so I would have lost all intuition as to the result. For simple cases, the counting approach is much easier and shows why the result is what it is.

This argument is at the heart of Steven Stogatz’s excellent article “Chances Are”, online at the New York Times (thanks for the pointer, Matt). He gives some excellent examples of conditional probability, including a great riff on the O.J. Simpson murder trial.

The prosecution spent the first 10 days of the trial introducing evidence that O.J. had a history of violence toward his ex-wife, Nicole. He had allegedly battered her, thrown her against walls and groped her in public, telling onlookers, “This belongs to me.” But what did any of this have to do with a murder trial? The prosecution’s argument was that a pattern of spousal abuse reflected a motive to kill. As one of the prosecutors put it, “A slap is a prelude to homicide.”

Alan Dershowitz countered for the defense, arguing that if even the allegations of domestic violence were true, they were irrelevant and should therefore be inadmissible. He later wrote, “We knew we could prove, if we had to, that an infinitesimal percentage — certainly fewer than 1 of 2,500 — of men who slap or beat their domestic partners go on to murder them.”

In effect, both sides were asking the jury to consider the probability that a man murdered his ex-wife, given that he previously battered her. But as the statistician I. J. Good pointed out, that’s not the right number to look at.

The real question is: What’s the probability that a man murdered his ex-wife, given that he previously battered her and she was murdered by someone? That conditional probability turns out to be very far from 1 in 2,500.

Turns out that probability is about 90%.

We are in the midst of a curriculum review at the Tepper School of Business and are considering what we absolutely have to be sure our MBAs know. Being able to work with conditional probability is very high on my list: it is an area where your intuition will almost surely lead you astray. And, as the Strogatz article points out, while Bayes Rule may get you the right results, simple counting arguments are much more convincing.