10s.
0d.: but we also impress upon him that in order to justify this statement he must commence to toss at least 1024 times, for in no less number can all the contingencies of gain and loss be exhibited and balanced. If he proposes to reach an average gain of £20, he will require to be prepared to go up to 39 throws, To justify this payment he must commence to throw 239 times,
i.e.
about a million million times. Not before he has accomplished this will he be in a position to prove to any sceptic that this is the true average value of a ‘turn’ extending to 39 successive tosses.
Of course if he elects to toss to all eternity we must adopt the line of explanation which alone is possible where questions of infinity in respect of number and magnitude are involved. We cannot tell him to pay down ‘an infinite sum,’ for this has no strict meaning. But we tell him that, however much he may consent to pay each time runs of heads occur, he will attain at last a stage in which he will have won back his total payments by his total receipts. However large n may be, if he perseveres in trying 2n times he may have a true average receipt of 1/2 (n + 1) pounds, and if he continues long enough onwards he will have it.
The problem will recur for consideration in a future chapter.
[1] The following statistics will give a fair idea of the wide range of experience over which such regularity is found to exist: “As illustrations of equal amounts of fluctuation from totally dissimilar causes, take the deaths in the West district of London in seven years (fluctuation 13.66), and offences against the person (fluctuation 13.61); or deaths from apoplexy (fluctuation 5.54), and offences against property, without violence (fluctuation 5.48); or students registered at the College of Surgeons (fluctuation 1.85), and the number of pounds of manufactured tobacco taken for home consumption (fluctuation 1.89); or out-door paupers (fluctuation 3.45) and tonnage of British vessels entered in ballast (fluctuation 3.43), &c.” [Extracted from a paper in the Journal of the Statistical Society, by Mr Guy, March, 1858; the ‘fluctuation’ here given is a measure of the amount of irregularity, that is of departure from the average, estimated in a way which will be described hereafter.]
[2] Transactions of the Cambridge Philosophical Society, Vol. IX.
p. 605. Reprinted in the collected edition of his writings, p. 50.