Those Average Speeds
I’ve been giving more thought to the “average speed camera” concept that I described in my last post. For those who haven’t read it, the concept is superficially simple: in the UK a car is timed entering a section of road at the point A and again leaving it at the point B. The distance between A and B has been measured and the car’s average speed is calculated. If the average speed is greater than the speed limit then the motorist is prosecuted because, somewhere between A and B he must have exceeded the speed limit. This is a rather subtle existence proof and I have been thinking about the exchange in court.
Sir Ethelred Carrotcake Q.C.: Tell me, where exactly was my client when the alleged offence took place?
P.C. Fred Smith: Well, I don’t actually know but it was somewhere between A and B.
Q.C.: And how far is it between A and B?
P.C.: Six miles.
Q.C.: Six miles! So you are saying that the alleged offence took place somewhere within a six mile section of road but you cannot say where. And how fast was my client alleged to have been driving?
P.C.: I don’t know, but it must have been more than the speed limit: 60 miles per hour.
Q.C.: You don’t know! So are you expecting the jury to convict my client because you think he may have driven at some speed greater than the speed limit (but don’t know by how much) at some point between A and B (but you don’t know where)?
P.C.: But his average speed was 70 miles per hour! So he must have been doing more than 60 somewhere.
Q.C.: And what makes you say that?
P.C.: But it’s obvious.
Q.C.: [turning to the jury] Members of the jury, I will be calling 10 expert witnesses this afternoon, all pure mathematicians, who will dazzle you with theorems and convince you that, far from being obvious, this law apparently relies on a very subtle mathematical theorem. But, let us test this P.C.’s understanding of the obviousness of average speeds. [turning to the witness] Now, let us assume for the sake of argument that the speed limit for the first three miles of this road was 60 miles per hour and for the second three miles 40 miles per hour. At what average speed would my client have had to travel before getting into trouble with you?
P.C.: 50 miles per hour, of course: the average of 60 and 40.
Q.C.: Would it surprise you to learn that all ten of my expert witnesses will say that the correct answer is 48?
P.C.: That can’t be true….
Q.C.: It certainly can’t be true if your simple idea of average speed is correct, can it? But [turning to the jury], let me summarise where we stand at the end of this witness’ evidence: he cannot say where my client is alleged to have committed this offence, he was not present when the alleged offence took place, he cannot say what offence my client is alleged to have committed since he makes no pretence to know the speed at which my client is alleged to have been travelling and he has a completely erroneous understanding of the concept of average speed, the very concept on which he is basing this flimsy case.
At this point the judge, convinced by Sir Ethelred’s argument and fearing the parade of mathematical expert witnesses, stopped the case and directed the jury to find the accused innocent so we will never know what the final outcome would have been.