Nonplussed! by Julian Havil Read Free Book Online Page B

difficult to decide. Divide these contacts equally, and we have 1,218 1/2 to 3,204 for the ratio of 6 to 5Pi, presuming that the greatness of the number of trials gives something near to the final average, or result in the long run: this gives Pi=3.1553. If all the 11 contacts had been treated as intersections, the result would have been Pi=3.1412, exceedingly near. A pupil of mine made 600 trials with a rod of the length between the seams, and got Pi=3.137. This method will hardly be believed until it has been repeated so often that ‘there never could have been any doubt about it.’
    We will look into this peculiar phenomenon, but first we will mention some related games.
    Fairground Games
    The study of ‘geometric probability’, where probabilities are determined by comparison of measurements, seems to have had its birth in 1777 (as did the greatest of all mathematicians, Gauss) in the paper, ‘Sur le jeu de franc-carreau’, published by Georges Louis Leclerc, Comte de Buffon. The game of throwinga small coin (‘ un ecu ’) onto a square grid was a popular pastime and the question of a fair fee to play the game naturally arose; put another way, what is the probability that the coin lands wholly in a square tile (‘ à franc-carreau ’)?

    Figure 7.1. The coin on the square
    Buffon correctly argued that the coin would land entirely within a square tile whenever the centre of the coin landed within a smaller square, whose side was equal to the side of a grid square less the diameter of the coin, as we see in figure 7.1 .
    If the grid square is of side a and the coin has diameter d (which we will suppose is not greater than), this means that, if we write this probability as p , we have

    where.
    For the game to be fair, the expected value of the game must be 0 and so, if it costs 1 unit to play and we are given w units if we win,

    which gives

    A plot of w against d / a is given in figure 7.2 .

    Figure 7.2. Winning behaviour

    Figure 7.3. The coin and ruled lines
    To entice the player to double their money, a simple calculation shows thator a little less than this if we are to make a profit!
    Moving from a square grid to sets of parallel lines makes the calculation even easier. If the lines are a constant distance h apart and the disc has a diameter d , it is clear from figure 7.3 that the disc will land within a pair of lines if its centre lies in a band of width h − d and so the probability that this happens is

    Our fair game would now force

    where d < h .

    Figure 7.4. Winning behaviour
    A plot of w against d / h is given in figure 7.4 .
    Another simple calculation shows that, to double the stake, d / h = 1/2.
    So far, these are geometric probabilities calculated in a reasonable manner to give reasonable answers. Now we move to the already heralded, seemingly simpler, but far more intriguing, case.
    Buffon’s (Short) Needle
    Buffon raised the question of throwing not a circular object, but an object of a different shape, such as a square, or a ‘baguette’ (a rod or stick), or, as he points out, ‘On peut jouer ce jeu sur un damier avec une aiguille à coudre ou une épingle sans tête.’ (‘You can play this game on a chequerboard with a sewing-needle or a pin without a head.’) It is said that he threw a classic French baguette over his shoulder onto a boarded floor to demonstrate a version of the idea. We come, then, to the phenomenon now universally known as Buffon’s Needle: if we throw a needle of length l on a board ruled with parallel lines, distance d (≥ l ) apart, what is the probability that the needle crosses one of the lines?
    In the eighteenth and nineteenth centuries such experiments were common, with probability considered as something of an experimental science. We have seen De Morgan detail the efforts of Mr Ambrose Smith of Aberdeen; this and the efforts of De Morgan himself are included in a table in the 1960 article, Geometricprobability and the number π ( Scripta Mathematica