Monday, October 25, 2010

How much information is stored in a lapchart?

If your sleep was somewhat disturbed over the weekend, it may have been a consequence of getting up in the middle of the night to watch the latest round of the Ryanair Formula 1 World Championship, hosted, it appears, at a South Korean fishing village. Equally likely, however, your mind may have been immedicably troubled by the question of how much information is encoded in a typical Grand Prix lapchart.

So, to soothe your passage Lethewards tonight, let us endeavour to address the latter source of vexation, at least.

Begin by supposing that there are 24 cars, and that a Grand Prix consists of 70 laps. For simplicity, let us also suppose that all 24 cars complete all 70 laps. There are 24! (twenty-four factorial) permutations of ('ways of ordering') the 24 cars on each lap. Assuming that the permutations on each lap are independent, (even if the reality is that they are highly correlated), gives us the following number of possible lapcharts:

(24!)70 ≈ 101665 .

Now, to find the amount of information, in bits, stored by a particular lapchart one merely has to take the log to the base 2 of the total number of possible lapcharts:

log2 (101665) ≈ 5,000 bits .

Dividing by the number of bits in a byte (eight), gives the number of bytes as approximately 625. In other words, there is, at most, about half a kilo-byte of information in a typical Grand Prix lapchart. Taking into account the average degree of correlation between the order on successive laps in modern Grand Prix racing would reduce this quantity quite considerably.

Sleep well.

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