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]--