ONT Re: Inquiry Driven Systems
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Note 15
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CP 2.653. Three Logical Sentiments (cont.)
| These considerations appear, at first sight, to dispose of
| the difficulty mentioned. Yet the case of the other side
| is not yet exhausted. 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.
| Taking all his risks collectively, then, it cannot be certain
| that they will not fail, and his case does not differ, except
| in degree, from the one last supposed. It is an indubitable
| result of the theory of probabilities that every gambler, if
| he continues long enough, must ultimately be ruined. Suppose
| he tries the martingale, which some believe infallible, and
| which is, as I am informed, disallowed in the gambling-houses.
| In this method of playing, he first bets say $1; if he loses
| it he bets $2; if he loses that he bets $4; if he loses that
| he bets $8; if he then gains he has lost 1 + 2 + 4 = 7, and
| he has gained $1 more; and no matter how many bets he loses,
| the first one he gains will make him $1 richer than he was in
| the beginning. In that way, he will probably gain at first;
| but, at last, the time will come when the run of luck is so
| against him that he will not have money enough to double, and
| must, therefore, let his bet go. This will 'probably' happen
| before he has won as much as he had in the first place, so that
| this run against him will leave him poorer than he began; some
| time or other it will be sure to happen. It is true that there
| is always a possibility of his winning any sum the bank can pay,
| and we thus come upon a celebrated paradox that, though he is
| certain to be ruined, the value of his expectation calculated
| according to the usual rules (which omit this consideration) is
| large. But, whether a gambler plays in this way or any other,
| the same thing is true, namely, that if [he] plays long enough
| he will be sure some time to have such a run against him as
| to exhaust his entire fortune. The same thing is true of an
| insurance company. Let the directors take the utmost pains to
| be independent of great conflagrations and pestilences, their
| actuaries can tell them that, according to the doctrine of
| chances, the time must come, at last, when their losses will
| bring them to a stop. They may tide over such a crisis by
| extraordinary means, but then they will start again in a
| weakened state, and the same thing will happen again all the
| sooner. An actuary might be inclined to deny this, because
| he knows that the expectation of his company is large, or
| perhaps (neglecting the interest upon money) is infinite.
| But calculations of expectations leave out of account the
| circumstance now under consideration, which reverses the
| whole thing. However, I must not be understood as saying
| that insurance is on this account unsound, more than other
| kinds of business. All human affairs rest upon probabilities,
| and the same thing is true everywhere. If man were immortal
| he could be perfectly sure of seeing the day when everything in
| which he had trusted should betray his trust, and, in short, of
| coming eventually to hopeless misery. He would break down, at last,
| as every great fortune, as every dynasty, as every civilization does.
| In place of this we have death.
|
| C.S. Peirce, 'Collected Papers', CP 2.653
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