## Sunday, August 16, 2009

### Frentzen, Schumacher, Villeneuve and probabilities At the Formula 1 World Championship finale in 1997, the two drivers fighting for the championship, Jacques Villeneuve and Michael Schumacher, both set exactly the same qualifying time, 1m 21.072. Not only that, but Villeneuve's team-mate, Heinz-Harald Frentzen, also set a time of exactly 1m 21.072. How improbable was that?

Well, not as improbable as one might think. Ian Stewart, writing with Jack Cohen and Terry Pratchett in The Science of Discworld (1999), points out that these three drivers could have been expected to set a time within a tenth of each other. The timing resolution was to thousandths of a second, hence there were effectively a hundred different times available to each of these drivers. Thus, the probability of an individual driver setting a specific time, such as A = 1m 21.072, was:

P(A) = 1/100

The probability of a conjunction of independent events is obtained by multiplying the probabilities of the individual events, so, as a first stab, one might guess that the probability of all three drivers setting the same time is as follows:

P(A&B&C) = 1/100 * 1/100 * 1/100 = 1/1,000,000 (i.e., one in a million)

This, however, is the probability of all three drivers setting a specific time such as 1m 21.072. It is not the specific time which is relevant here, but the probability of all three drivers setting the same time. Once the first driver of the triumvirate had set a time of 1m 21.072, we want to know the conditional probability of the second and third drivers also setting that time. To obtain the conditional probability of A&B&C, given the occurrence of A, one simply divides the probability of A&B&C by the probability of A:

P(A&B&C)/P(A) = 1/10,000

So the probability of Frentzen, Schumacher and Villeneuve setting the same time to a thousandth of a second, was one-in-ten-thousand. Still a long shot, but hardly astronomical odds. There have been in the order of a thousand World Championship Grands Prix since the inauguration of the championship in 1950, and although timing to thousandths of a second is a more recent innovation, we can say that the probability of three drivers setting exactly the same time over a thousand Grands Prix is one-in-ten.

In a ensemble consisting of ten separate planets within a galaxy, each hosting a thousand Grands Prix, and timing to thousandths of a second, one would expect there to be one planet on which three drivers set exactly the same time in at least one Grands Prix.

To expect that it happens in qualifying for the World Championship shoot-out? Well, once the existence of Bernie Ecclestone is factored into the equation, the improbability becomes much smaller again...