Odds

“ Suppose that all the men, women and children on the face of the earth were to keep on tossing coins at the rate of a million a second for a million years, the odds would still be too great for us to realise against any one person succeeding in performing the above feat, and yet the number representing the odds would be only half as long as the one already given. ”

“ He always appears to incline to the opinion that the individual judgment in probability is to admit of justification; that when we say, for instance, that the odds in favour of some event are three to two, that we can explain and justify our statement without any necessary reference to a series or class of such events. ”

“ Many who do not bet themselves, find an interest in watching the varying fortunes of the horses which are held by the initiated to be leading favourites, or to fall into the second rank, or merely to have an outside chance of success. It is amusing to notice, too, how frequently the final state of the odds is falsified by the event ”

“ Each event or contingency must, of course, be uncertain, or, at all events, unknown to the parties, that is, it represents a chance, and where the chances are agreed to be uneven the inequality is represented by odds. ”

“ Suppose the newspapers inform us that the betting is 2 to 1 against a certain horse for such and such a race, what inference are we to deduce? To learn this let us conceive a case in which the true odds against a certain event are as 2 to 1. Suppose there276 are three balls in a bag, one being white, the others black. Then, if we draw a ball at random, it is clear that we are twice as likely to draw a black as to draw a white ball. ”

“ If one is satisfied with smaller odds, and with better chances of winning, one can place his money between two numbers, or in the middle of four numbers, or on a transversal of three numbers, or on a double transversal of six numbers, or in various other ways. ”

“ These odds are regulated upon a scale of equity, from which there can be no departure; it being an invariable principle of the game, that any person laying an unfair bet, or deviating from the fixed odds, can never win. ”

“ If the player gives odds, and yet omits to remove the odds from the board at the commencement, he may recommence the game, and remove the odds given, provided he discover his error before playing his fourth move. ”

“ But the chance of this occurrence is represented by a fraction whose numerator is 1, and denominator contains 77 figures, and is therefore utterly inappreciable by the imagination. It must be admitted, though, that there is something hypothetical about such an example, for we could not really know that the dies were fair with a confidence even distantly approaching such prodigious odds. ”

“ In a barrel of odds and ends it is different; things get mixed up, and the juice kind of swaps around, and the things go better. ”

“ Now, if several pieces of evidence, distinct in their origin, concur to a given conclusion, the probability that that conclusion is correct is far greater than what results from each alone; and mounts up soon to something much exceeding what bettors at races call certainty odds. However unversed the average man may be in calculating the probability, he recognizes the fact in his dealings with his fellows by the way he attaches weight to concurrent testimony. ”

“ Granting this axiom, and assuming that we have no ground for selecting any one kind as more likely than the rest to surpass the average proportion, it follows that we can not rationally presume this of any, which we should do if we were to bet in favor of it, receiving less odds than in the ratio of the number of the other kinds. ”

“ The experience however of a few games taught me that my discovery was not quite of so lucrative a nature as I had supposed. The odds did not every game vary, from side to side; people were not always inclined to bet the odds ”

“ "You don't go by odds. You know what's coming. You've seen the cards already." She studied his face intently. "No, you can't be cold-decked. It wouldn't be possible." ”

“ What's the odds betwixt the two? ”

“ Shrewd players, however, prefer some other main, with the view of having a more favourable chance to depend upon of winning both stake and odds. For example, let us reverse what was mentioned above, and suppose the caster to call 5 and throw 7; he then will have 7 as his chance to win with odds of 3 to 2 IN HIS FAVOUR.
'Such is the game of English Hazard, at which large fortunes have been won and lost. ”

“ Judging is, as it were, balancing an account, and determining on which side the odds lie. ”

“ On the following day, as MacNutt had arranged, the two paid their first visit to Penfield's lower house, from which Durkin carried away confused memories of a square-jawed door-keeper—who passed him, readily enough, at a word from the girl; of well-dressed men and over-dressed women crowded about a smoky, gas-lit room, one side of which was taken up with a black-board on which attendants were feverishly chalking down entries, jockeys, weights and odds, while on the other side of the room opened the receiving- and paying-tellers' little windows, through which now and then he saw hurrying clerks ”

“ According to the view here taken, these are the only fundamental rules for the calculation of chances. An additional one, derived from a different conception of probability, is given in some treatises, which if it be sound might be made the basis of a theory of reasoning. Being, as I believe it is, absolutely absurd, the consideration of it serves to bring us to the true theory; and it is for the sake of this discussion, which must be postponed to the next number, that I have brought the doctrine of chances to the reader's attention at this early stage of our studies of the logic of science. ”

“ Ninth Principle.—In a series of probable events of which the ones produce a benefit and the others a loss, we shall have the advantage which results from it by making a sum of the products of the probability of each favorable event by the benefit which it procures, and subtracting from this sum that of the products of the probability of each unfavorable event by the loss which is attached to it. If the second sum is greater than the first, the benefit becomes a loss and hope is changed to fear. ”

“ Fifth Principle.—If we calculate à priori the probability of the occurred event and the probability of an event composed of that one and a second one which is expected, the second probability divided by the first will be the probability of the event expected, drawn from the observed event.
Here is presented the question raised by some philosophers touching the influence of the past upon the probability of the future. Let us suppose at the play of heads and tails that heads has occurred oftener than tails. ”

“ The theory of chance consists in reducing all the events of the same kind to a certain number of cases equally possible, that is to say, to such as we may be equally undecided about in regard to their existence, and in determining the number of cases favorable to the event whose probability is sought. ”

“ Suppose they agree to leave off playing when one player, A, wants x “points” (games to be won) in order to complete the assigned number, while the second player wants x′ points: how ought they to divide the stakes? This is a question in Expectation, but its difficulty consists in determining the probability that one of the players, say A, shall win the stakes. Let that probability be yx,x′. Then, after the next game, if A has won, the probability of his winning the stakes will be yx-1,x′. But if A loses, B winning, the probability will be yx,x′-1. But these alternatives are equally likely. ”

“ But when the manner in which the mixing of the numbers in this lottery is considered, the past ought to have no influence upon the future. The very frequent drawings of a number are only the anomalies of chance; I have submitted several of them to calculation and have constantly found that they are included within the limits which the supposition of an equal possibility of the drawing of all the numbers allows us to admit without improbability. ”

“ The probability of events serves to determine the hope or the fear of persons interested in their existence. The word hope has various acceptations; it expresses generally the advantage of that one who expects a certain benefit in suppositions which are only probable. This advantage in the theory of chance is a product of the sum hoped for by the probability of obtaining it; it is the partial sum which ought to result when we do not wish to run the risks of the event in supposing that the division is made proportional to the probabilities. ”

“ Proceeding then in that reasoning, wherein we have advanc'd three steps; that chance is merely the negation of a cause, and produces a total indifference in the mind; that one negation of a cause and one total indifference can never be superior or inferior to another; and that there must always be a mixture of causes among the chances, in order to be the foundation of any reasoning: We are next to consider what effect a superior combination of chances can have upon the mind, and after what manner it influences our judgment and opinion. ”

“ Although probability will probably manifest its effect in, say, a thousand risks, by a certain proportion between the numbers of successes and failures, yet this, as we have seen, is only to say that it certainly will, at length, do so. Now the number of risks, the number of probable inferences, which a man draws in his whole life, is a finite one, and he cannot be absolutely certain that the mean result will accord with the probabilities at all. ”

“ What I have said concerning the probability of chances can serve to no other purpose, than to assist us in explaining the probability of causes; since 'tis commonly allow'd by philosophers, that what the vulgar call chance is nothing but a secret and conceal'd cause. That species of probability, therefore, is what we must chiefly examine. ”

“ But so also, for aught we knew, might there be more red balls than black and white, or more black balls than white and red, and in such case the effect of additional knowledge would be to prove to us that our bet was more advantageous than we had supposed it to be. There is in the existing state of our knowledge a rational probability of two to one against white; a probability fit to be made a basis of conduct. ”

“ The irregular events of nature are not exactly comparable to the drawing of the numbers of a lottery in which all the numbers are mixed at each drawing in such a manner as to render the chances of their drawing perfectly equal. The frequency of one of these events seems to indicate a cause slightly favoring it, which increases the probability of its next return, and its repetition prolonged for a long time, such as a long series of rainy days, may develop unknown causes for its change ”

“ The analysis of probabilities leads to this general theorem.
By the repetition of an advantageous event, simple or compound, the real benefit becomes more and more probable and increases without ceasing; it becomes certain in the hypothesis of an infinite number of repetitions; {150} and dividing it by this number the quotient or the mean benefit of each event is the mathematical hope itself or the advantage relative to the event. It is the same with a loss which becomes certain in the long run, however small the disadvantage of the event may be. ”

“ They see only the possibility by a small stake of gaining a considerable sum, and the projects which their imagination brings forth, exaggerate to their eyes the probability of obtaining it; the poor man especially, excited by the desire of a better fate, risks at play his necessities by clinging to the most unfavorable combinations which promise him a great benefit. All would be without doubt surprised by the immense number of stakes lost if they could know of them ”

“ The mind has its illusions as the sense of sight; and in the same manner that the sense of feeling corrects the latter, reflection and calculation correct the former. Probability based upon a daily experience, or exaggerated by fear and by hope, strikes us more than a superior probability but it is only a simple result of calculus. Thus we do not fear in return for small advantages to expose our life to dangers much less improbable than the drawing of a quint in the lottery of France ”

“ Note, too, how in the holder of a lottery-ticket hope generates a belief utterly at variance with probability as numerically estimated, and how an excited inventor has a confidence in success which calm judges see to be impossible. ”