Then the odds in favour of the assertion are IA x e' x x (1—v) x (1—v) . . . . to (1—p) x (1 — ,u') x . x v x v' .... Let the pro bability of the assertion thence obtained (the first of the two divided by their sum) be called at : and remember that the receiver himself is one of those witnesses, as before explained.
Let a slumber of arguments of which the several probabilities of validity (that is, of their proving their conclusions true) be a, a', .... The probability that the conclusion is proved—that is, that one or more are valid—is 1—(I—a) (1—a) .... If arguments be produced for the contradictory conclusion, of validities b, , &c., then (1-0 . . . . is the probability that the conclusion is disproved. We speak of separate cases the combination of arguments for and against will be instantly seen. Let these results be A and 11. We may treat the conclusion, as now asserted to an unbiassed mind by a single witness of credibility am, supported by ouc argument of validity A, and opposed by one argurnent of validity as. The only difference is that if there be a plurality of witnesses or of arguments, we have to calculate At, s, n; which, when there is one only of each kind, are supposed given.
When testimony and argument are combined, the odds in favour of the ,conclusion aro at (1—n), to (1-11) (1— a). The following cases may be distinguished:— 1. No evidence; arguments on both sides. Hero at= and the odds: are to 1—e. When A and n are very near to unity, or the urn lies between that of 418-10 to 182+10 and 418r 10 to 182-10, or that of 1000 to 471 and 1000 to 4021 Here a=418, 6..182. km10; the square root above mentioned has 1•2021 for its logarithm, and 10 divided by this square root gives -63 nearly. And A being -63, 11 is also -63 very nearly, which is the probability required; or It is 63 to 37, or nearly 2 to 1, that for every 1000 white balls in the urn, there are between 402 and 471 black ones.
Prom.= 4.—When r happens much oftener than q, we feel a very high degree of probability (moral certainty) that the same thing would happen in the long run, or would continua how long soever we might u on trying, unless there should arrive some change of circumstances. But when r does not happen much oftener than q, we do not feel the same degree of assurance; for though Q might really happen oftener than r in the long run, yet the casual fluctuations of events might make the contrary appear in any one set of trials. Suppose then that
In a+6 trials, e has happened a times, and q has happened 6 times, a aril 6 being nearly equal : what presumption can thence be derived that the excess will be on the same side in the long ran on which it is in the a + 6 trials! Hum—Divide the difference of a and 6 by the square root of twice their sum : let the result be A, and find the corresponding B; to this add 1, and divide by 2 : the result is the probability required.
Ezell ri.e.—Buffon, in a particular experiment, had to throw a coin (ray a halfpenny) 4140 times ; the result was 2048 heads and 2092 tails : what is the probability that some excess of tail over head would have continued for ever; that is, that the coin he used had in its con atruction some mechanical tendency to fall tail rather than head! Mere a=2045, 6=-2092 ; the difference divided by the square root of twice the sum is -48, which being A, B is -50, and 1.50 divided by 2 is the probability required. This is 1, so that it is 3 to 1 that some excess or other of tails would be found in the long run. Observe how ever that this does not mean an excess to the amount of 44 in every 4140 trials ; but only some excess, small or great : observe also that the character of the mint and its processes are supposed wholly unknown, so that no conclusion can be formed about the character of the coiu except from the observed event.
The experiment of Buffon, and its particular object, will be found in the' Essay on Probabilities and Life Contingencies,' in the Cabinet Cycloptedia ; ' in which work will also be found a larger collection of such problems as the preceding, with fuller explanation and more extensive tables. These problems have been here introduced that the reader may have an opportunity of comparing and verifying the general sameness of results which follows from the two (perhaps) apparently differing notions from which the idea of the measure of probability may be derived.