§2. The Meaning of Probability
21. This is the problem of insurance. Now in order that probability may have any bearing on this problem, it is obvious that it must be of the nature of a real fact and not a mere state of mind. For facts only enter into the solution of the problem of insurance. And this fact must evidently be a fact of statistics.
Without now going into certain reasons of detail that I should enter into if I were lecturing on probabilities, it must be that probability is a statistical ratio; and further, in order to satisfy still more special conditions, it is convenient, for the class of problems to which insurance belongs, to make it the statistical ratio of the number of experiential occurrences of a specific kind to the number of experiential occurrences of a generic kind, in the long run.1)
In order, then, that probability should mean anything, it will be requisite to specify to what species of event it refers and to what genus of event it refers.
It also refers to a long run, that is, to an indefinitely long series of occurrences taken together in the order of their occurrence in possible experience.
In this view of the matter, we note, to begin with, that a given species of event considered as belonging to a given genus of events does not necessarily have any definite probability. Because [it may be the case that] the probability is the ratio of one infinite multitude to another. Now infinity divided by infinity is altogether indeterminate, except in special cases.
22. It is very easy to give examples of events that have no definite probability. If a person agrees to toss up a cent again and again forever and beginning as soon as the first head turns up whenever two heads are separated by any odd number of tails in the succession of throws, to pay 2 to that power in cents, provided that whenever the two successive heads are separated by any even number of throws he receives 2 to that power in cents,•P1 it is impossible to say what the probability will be that he comes out a winner. In half of the cases after the first head the next throw will be a head and he will receive (-2)0 = 1 cent. Which since it happens half the time will be in the long run a winning of 1/2 a cent per head thrown.
But in half of the other half the cases, that is in 1/4 of all the cases, one tail will intervene and he will have to receive (-2)1 = -2 cents, i.e., he will have to pay 2 cents, which happening 1/4 of the time will make an average loss of 1/2 a cent per head thrown.
But in half the remaining quarter of the cases, i.e., of all the cases, two tails will intervene and he will receive (-2)2 = 4 cents which happening one every eight times will be worth 1/2 a cent per head thrown and so on; so that his account in the long run will be 1/2-1/2+1/2-1/2+1/2-1/2+1/2-1/2 ad infinitum, the sum of which may be 1/2 or may be zero. Or rather it is quite indeterminate.
If instead of being paid (-2) n when n is the number of intervening tails, he were paid (-2) n2 the result would be he would probably either win or lose enormously without there being any definite probability that it would be winning rather than losing.
I think I may recommend this game with confidence to gamblers as being the most frightful ruin yet invented; and a little cheating would do everything in it.
23. Now let us revert to our original problem1) and consider the state of things after every other bet. After the second, 1/4 of the players will have gained, gone out, and been replaced by players who have gained and gone out, so that a number of francs equal to half the number of seats will have been paid out by the bank, 1/4 of the players will have gained and gone out and been replaced by players who have lost, making the bank even; 1/4 of the players will have lost and then gained, making the bank and them even; 1/4 of the players will have lost twice, making a gain to the bank of half as many francs as there are seats at the table. The bank then will be where it was. Players to the number of three-quarters of the seats will have netted their franc each; but players to the number of a quarter of the seats will have lost two francs each and another equal number one franc each, just paying for the gains of those who have retired.
That is the way it will happen every time.
Just before the fifth bet of the players at the table, 3/8 will have lost nothing, 1/4 will have lost one franc, 1/4 two francs, 1/16 three francs and 1/16 four francs. Thus some will always have lost a good deal. Those who sit at the table will among them always have paid just what those who have gone out have carried away.
24. But it will be asked: How then can it happen that all gain? I reply that I never said that all would gain, I only said that the probability was 1 that anyone would ultimately gain his franc. But does not probability 1 mean certainty? Not at all, it only means that the ratio of the number of those who ultimately gain to the total number is 1. Since the number of seats at the table is infinite the ratio of the number of those who never gain to the number of seats may be zero and yet they may be infinitely numerous. So that probabilities 1 and 0 are very far from corresponding to certainty pro and con.2)
