been attempted[5] to explain the preponderance of male births over female by assuming that the chances of the two are equal, but that the general desire to have a male heir tends to induce many unions to persist until the occurrence of this event, and no longer. It is supposed that in this way there would be a slight preponderance of families which consisted of one son only, or of two sons and one daughter, and so forth.
This is quite fallacious (as had been noticed by Laplace, in his Essai); and there could not be a better instance chosen than this to show just what we can do and what we cannot do in the way of altering the luck in a real chance-succession of events. To suppose that the number of actual births could be influenced in the way in question is exactly the same thing as to suppose that a number of gamblers could increase the ratio of heads to tails, to something over one-half, by each handing the coin to his neighbour as soon as he had thrown a head: that they have only to leave off as soon as head has appeared; an absurdity which we need not pause to explain at this stage. The essential point about the ‘Martingale’ is that, whereas the occurrence of the events on which the stakes are laid is unaffected, the stakes themselves can be so adjusted as to make the luck swing one way.
§ 18. In the second place, this example brings before us what has had to be so often mentioned already, namely, that the series of Probability are in strictness supposed to be interminable. If therefore we allow either party to call upon us to stop, especially at a point which just happens to suit him, we may get results decidedly opposed to the integrity of the theory. In the case before us it is a necessary stipulation for A that he may be allowed to leave off when he wishes, that is at one of the points at which the throw is in his favour. Without this stipulation he may be left a loser to any amount.
Introduce the supposition that one party may arbitrarily call for a stoppage when it suits him and refuse to permit it sooner, and almost any system of what would be otherwise fair play may be converted into a very one-sided arrangement. Indeed, in the case in question, A need not adopt this device of doubling the stakes every time he loses. He may play with a fixed stake, and nevertheless insure that one party shall win any assigned sum, assuming that the game is even and that he is permitted to play on credit.
§ 19. (V.) A common mistake is to assume that a very unlikely thing will not happen at all. It is a mistake which, when thus stated in words, is too obvious to be committed, for the meaning of an unlikely thing is one that happens at rare intervals; if it were not assumed that the event would happen sometimes it would not be called unlikely, but impossible. This is an error which could scarcely occur except in vague popular misapprehension, and is so abundantly refuted in works on Probability, that it need only be touched upon briefly here. It follows of course, from our definition of Probability, that to speak of a very rare combination of events as one that is ‘sure never to happen,’ is to use language incorrectly. Such a phrase may pass current as a loose popular exaggeration, but in strictness it involves a contradiction. The truth about such rare events cannot be better described than in the following quotation from De Morgan:[6]—
“It is said that no person ever does arrive at such extremely improbable cases as the one just cited [drawing the same ball five times running out of a bag containing twenty balls]. That a given individual should never throw an ace twelve times running on a single die, is by far the most likely; indeed, so remote are the chances of such an event in any twelve trials (more than 2,000,000,000 to 1 against it) that it is unlikely the experience of any given country, in any given century, should furnish it. But let us stop for a moment, and ask ourselves to what this argument applies. A person who rarely touches dice will hardly believe that doublets sometimes occur three times running; one who handles them frequently knows that such is sometimes the fact. Every very practised user of those implements has seen still rarer sequences. Now suppose that a society of persons had thrown the dice so often as to secure a run of six aces observed and recorded, the preceding argument would still be used against twelve. And if another society had practised long enough to see twelve aces following each other, they might still employ the same method of doubting as to a run of twenty-four; and so on, ad infinitum. The power of imagining cases which contain long combinations so much exceeds that of exhibiting and arranging them, that it is easy to assign a telegraph which should make a separate signal for every grain of sand in a globe as large as the visible universe, upon the hypothesis of the most space-penetrating astronomer. The fallacy of the preceding objection lies in supposing events in number beyond our experience, composed entirely of sequences such as fall within our experience. It makes the past necessarily contain the whole, as to the quality of its components; and judges by samples. Now the least cautious buyer of grain requires to examine a handful before he judges of a bushel, and a bushel before he judges of a load. But relatively to such enormous numbers of combinations as are frequently proposed, our experience does not deserve the title of a handful as compared with a bushel, or even of a single grain.”
§ 20. The origin of this inveterate mistake is not difficult to be accounted for. It arises, no doubt, from the exigencies of our practical life. No man can bear in mind every contingency to which he may be exposed. If therefore we are ever to do anything at all in the world, a large number of the rarer contingencies must be left entirely out of account. And the necessity of this oblivion is strengthened by the shortness of our life. Mathematically speaking, it would be said to be certain that any one who lives long enough will be bitten by a mad dog, for the event is not an impossible, but only an improbable one, and must therefore come to pass in time. But this and an indefinite number of other disagreeable contingencies have on most occasions to be entirely ignored in practice, and thence they come almost necessarily to drop equally out of our thought and expectation. And when the event is one in itself of no importance, like a rare throw of the dice, a great effort of imagination may be required, on the part of persons not accustomed to abstract mathematical calculation, to enable them to realize the throw as being even possible.
Attempts have sometimes been made to estimate what extremity of unlikelihood ought to be considered as equivalent to this practical zero point of belief. In so far as such attempts are carried out by logicians, or by those who are unwilling to resort to mathematical valuation of chances, they must be regarded as merely a special form of the modal difficulties discussed in the last chapter, and need not therefore be reconsidered here; but a word or two may be added concerning the views of some who have looked at the matter from the mathematician's point of view.
The principal of these is perhaps Buffon. He has arrived at the estimate (Arithmétique Morale § VIII.)
that this practical zero is equivalent to a chance of 1/10,000. The grounds for selecting this fraction are found in the fact that, according to the tables of mortality accessible to him, it represents the chance of a man of 56 dying in the course of the next day. But since no man under common circumstances takes the chance into the slightest consideration, it follows that it is practically estimated as having no value.