Free 100 Essential Things You Didn't Know You Didn't Know by John D. Barrow Page A

In year two, if the rainfall is independent of what it was in year 1, then it has a chance of 1/2 of being a record by beating the year-1 rainfall and a chance of 1/2 of not beating the year-1 rainfall. So the number of record years we expect in the first two years is 1 + ½. In year 3 there are just two ways in which the six possible rankings (i.e. a 1 in 3 chance) of the rainfall in years 1, 2 and 3 could produce a record in year 3. So the expected number of record years after 3 years of record keeping is 1 + ½ + ⅓. If you keep on going, applying the same reasoning to each new year, you will find that after n independent years of gathering data, the expected number of record years is the sum of a series of n fractions:
    1 + ½ + ⅓ + ¼ + . . . + 1/n
    This is a famous series that mathematicians call the ‘harmonic’ series. Let’s label as H(n) its sum after n terms are totalled; so we see that H(1) = 1, H(2) = 1.5, H(3) = 1.833, H(4) = 2.083, and so on. The most interesting thing about the sum of this series is that it grows so very slowly as the number of terms increases, fn1 so H(256) = 6.12 but H(1,000) is only 7.49 and H(1,000,000) = 14.39.
    What does this tell us? Suppose that we were to apply our formula to the rainfall records for some place in the UK from 1748 to 2004 – a period of 256 years. Then we predict that we should find only H(256) = 6.12, or about 6 record years of high (or low) rainfall. If we look at the rainfall records kept by Kew Gardens for this period then this is the number of record years there have been. We would have to wait for more than a thousand years to have a good chance of finding even 8 record years. Records are very rare if events occur at random.
    In the recent past there has been growing concern around the world about the evidence for systematic changes in climate, so called ‘global warming’, and we have noticed an uncomfortably large number of local climatic records in different places. If new records become far commoner than the harmonic series predicts, then this is telling us that annual climatic events are no longer independent annual events but are beginning to form part of a systematic non-random trend.
    fn1 In fact, when n gets very large H(n) increases only as fast as the logarithm of n and is very well approximated by 0.58 + ln(n).

29
    A Do-It-Yourself Lottery
    The Strong Law of Small Numbers
: There are not enough small numbers to satisfy all the demands placed upon them.
    Richard Guy
    If you are in need of a simple but thought-provoking parlour game to keep guests amused for a while, then one you might like to try is something that I call the Do-It-Yourself Lottery. You ask everyone to pick a positive whole number, and write it on a card along with their name. The aim is to pick the smallest number
that is not chosen by anyone else
. Is there a winning strategy? You might think you should go for the smallest numbers, like 1 or 2. But won’t other people think the same, and so you won’t end up with a number that is not chosen by someone else. Pick a very large number – and there are an infinite number of them to choose from – and you will surely lose. It’s just too easy for someone else to pick a smaller number. This suggests that the best numbers are somewhere in between. But where? What about 7 or 11? Surely no one else will think of picking 7?
    I don’t know if there is a winning strategy, but what the game picks up on is our reluctance to think of ourselves as ‘typical’. We are tempted to think that we could pick a low number for some reason that no one else will think of. Of course, the reason why opinion polls can predict how we will vote, what we will buy, where we will go on holiday, and how we will respond to an increase in interest rates is precisely because we are all so similar.
    I have another suspicion about this game. Although there is an infinite collection of numbers to choose from, we forget about most of them. We set a