The Abnormal Distribution

Paul Marks of the New Scientist has a couple of good recent articles on the volcanic-ash problem for commercial aviation, one from today and one from last week.

I talked about a simple calculation of this risk in my Risk course this morning, since it is topical, it shows practical issues well, and it fits in about an hour’s lecturing (with anecdotes). It seems that few people want to or can perform an elementary risk calculation about flying in the volcanic ash from Eyjafjallajökull. Here goes. It’s very crude, but still leads to some insight.

Let us classify first the outcome categories per flight. I choose four:

1. No damage
2. Engine needs thorough inspection and cleaning
3. Engine needs major overhaul
4. Engines stop in flight.

All of these have happened. 1 to the majority of recent airline flights, 2 to a couple of Ryanair planes, and to the Finnish F-18s that had an encounter on April 15 , the day before the first ban, reported here previously, 3 to the (in)famous NASA DC-8
(at a cost of $3.2m, so one reads), 4 to Eric Moody on the famous BA 747 in 1982.

One can almost directly read off the severity from these. Let us consider units to be equivalently pounds or euros or dollars. The sign “^” means “to the”, the exponential. So, e.g., 10^4 = 10,000, 10^6 = 1,000,000.

Severity of events (event classes) 1-4
1. 0
2. 10^4 to 10^5
3. 10^6 to 10^7
4. If a catastrophe is caused (i.e. the airplane does not succeed in making a dead-stick landing on an airport) then 10^8-10^9

It is curious that these four categories fit so crudely but neatly into powers of 10, covering the range.

So the risk is (the old De Moivre definition from 1711):

probability(1).severity(1) + probability(2).severity(2) + probability(3).severity(3) + probability(4).severity(4)

In fact, this is only a crude estimate of severity, since if some engine is found to be damaged, then all engines on all airplanes flying into or from those airports that engine flew into and around those routes that engine took will have to be inspected as well, and that might run into the hundreds. This calculation does not take account of these knock-on effects.

Using severity(1) = 0, the risk per flight then lies between

10^4 x prob(2) + 10^6 x prob(3) + 10^8 x prob(4)
and
10^5 x prob(2) + 10^7x prob(3) + 10^9 x prob(4)
(using the factors of ten associated with the severity ranges)

Consider your average intraeuropean flight, say Air Berlin flying Paderborn-London Stansted. Boeing 737NG, let’s say 150 people on board (this is an overestimate), paying €100 per seat (actually, it’s lower, and much of that is airport tax). Your revenue for the flight is at most €15,000 (and a lot less if you take out airport tax). So your expected value of loss, the risk, above, must be less than this if you hope to do better than by not flying. So your decision criterion is

10^4 x prob(2) + 10^6 x prob(3) + 10^8 x prob(4) < 15,000

if you take the lower estimate of risk, and

10^5 x prob(2) + 10^7 x prob(3) + 10^9 x prob(4) < 15,000, that is
10^4 x prob(2) + 10^6 x prob(3) + 10^8 x prob(4) < 1,500

if you take the higher.

Let us take the lower estimate. You can handle a cleaning event without much trouble, but you had better be sure, to break even, that you have at most one chance in just over 60 flights of an overhaul event, and only one chance in just over 6,000 flights of an engine-out event.

Given what was known on April 16th about outcomes (for example, that the Finnish engines might be trashed), I wonder how much of what we heard from airline chiefs complaining about not being able to fly was political manoeuvring for government handouts to “compensate” them for being forced to do what a risk analysis would have told them to do anyway?

PBL

The Times has written a blog-article on the proportion of the new UK government who went to Oxford (in fairness, I must point out that some proportion went to the Other Place, which is also rumored to be quite good). A perennial topic. I enjoyed reading the comments. But then I wondered whether the question could be seriously answered, and decided to have a go. (People may see the beginnings of an answer right there.)

My first degree: Oxford; my others: UC Berkeley (also a top tenner). I taught very briefly at Stanford, have worked at unis in Switzerland, France, Scotland and, for 15 years now, in Germany. I think I have a basis for comparison.

I felt like an outsider in Oxford, oppressed by the pressure of trying to achieve, and feeling that I wasn’t up to it, a feeling that it took me another decade to learn to ignore. But 40 years later, most of my matriculating class, including 5 of 7 maths people, turn up every few years for the reunions (one has died, and the Wykehamist disappeared during the course – is that what they learn there? Good prep for a career in offshore finance, I would think ). Two of us turn up for the Maths Institute Garden Party every so often. Contact with pals at UC Berkeley lasted longer, for I was there more than twice as long; but I make only occasional email contact with one or the other.

Last year, there was a reunion to celebrate 40 years of my Oxford degree course, in Maths and Philosophy, and lots of people turned up, including all those still alive – two of them octogenarians – who were responsible for setting it up, as well as all the holders over the decades of the associated Chair. There were more intense discussions over those two days than I have experienced at most conferences. It was my most delightful intellectual experience of the last decade (here, my heartfelt thanks to Hilary Priestley, Dan Isaacson, and Jochen Königsmann for organising it!).

Compare. My Bielefeld colleague, Ipke Wachsmuth, a delightful man whom everybody likes, just celebrated his 60th with a symposium and fun party at which he played lots of blues harmonica (about which I learnt that the hard part is picking the kit that fits the tune). Lots of people there, but just four of us Informatics faculty, out of fourteen. And the oldest of us is 60 (him). It ain’t the same as in Oxford.

So what are the factors? First, some commentators said “contacts”. It is more than that. The college system somehow fosters bonds of shared experience, which may well directly benefit those who go into banking, law, or politics, which are all about trust. (Unlike Maths, which is about being faster than the next guy or gal, and Philosophy, which is about calling other people idiots as politely as possible, a skill only half of which I learnt.)

In the case of those in my small degree program, it is also a matter of shared intellectual value – value fostered by the founders, John Lucas, Sir Michael Dummett, the late Robin Gandy; the first holders of the chair, Dana Scott, Angus Macintyre; and Robin’s successor Alex Wilkie. People at the very top of their field, world-wide, with whom undergrads like me could sit down to tea and discussion a few times a week. That doesn’t happen elsewhere to anywhere near the same extent. But maybe it is invalid to generalise from my degree program to all those at Oxford.

Second, the tutorial system of teaching is unique and structurally supports “thinking outside the box”, if that’s what you can and want to do. The work is much less routine than, say, handing in the weekly homework exercises for a Stanford course, and it is always demanding because tutors tailor it to you – they have to do so, to keep their own interest up. Ah, yes, those tutors who spend more time teaching fewer students than any academics anywhere else in the world. Thank you, people, for your devotion! In my case, especially Ian Macdonald, who encouraged my interest in logic and encouraged me to switch to the “right” degree course, the eccentric Mark Broido, who set the hardest problems but pointed out that the only person who really cared if I solved them was me, an important lesson for a 19-year-old expecting to be told what to do, and Ralph C.S. Walker, who talked me down for two hours after I had royally blown the first paper in Finals, thereby enabling me to do passably on the rest.

Finally, a more diffuse factor. Do I care that the Nobel-Memorial Econ went to UCB last year? A little bit, yes, as with all those other Nobel Prize/Fields Medal/Turing Award winners there. Do I care that 4 out of 24 current UK cabinet members, plus the Attorney General, went to my Oxford college, Magdalen? Yes, most definitely. I am quite proud, even though I know none of them. So, third, the system seems to foster pride in one’s notionally shared common experience. That is a main bonding mechanism in successful governments, isn’t it? And how many experiences in life foster that? Like it or loathe it, it could be a factor. It seems to happen in France, too.

BTW, I am far more Whig than Tory and I guess the cabinet’s now both – does one say Whory, or is that too rude?.
BTW, II, the Other Place is organised more or less the same way, so similar observations hold, but of course just not as well…..
BTW, III, someone pointed out that “the Other Place” should be capitalised. Maybe; I’ve done so. Sorry.

PBL