MW foot’ ON PROBABILITY. 49 
| deposit ought to be inversely proportional to the risk, which therefore he 
8 proposed to examine. The result of his examination was, that, in his opi- 
nion, a very small probability should be considered as none, and might be 
; entirely disregarded. He illustrates this, by supposing “ Peter to play with 
James, on this condition, that if a tossed halfpenny fall head, in the hun- 
po dredth toss, and not before, he is to receive from James 2100 crowns, in which 
n0minagig case the ordinary rule would determine Peter to give James one crown at 
TY one, thy the beginning of the game. I say, Peter ought not to give this crown, be- 
ried event cause he will lose it certainly,—because head will fall certainly before the 
0 the sup. 100th toss, although not necessarily.” Again, he says, ““ We must distin- 
' te paper guish between what is metaphysically, and what is physically possible. In 
Sof this jp. the first class is everything whose existence does not imply an absurdity; in 
1 ourselves, the second, everything whose existence not only does not imply an absurdity, 
tioundeq the but even implies nothing too extraordinary, and beyond the daily course of: 
Which hag events. It is melaphysically possible that two sixes may be thrown on two 
Pen again, dice a hundred times in succession; but it is impossible physically, because 
already sen it has never happened, and never will happen.” In the same memoir he 
- advances the opinion, that the oftener an event has already happened in 
em, has beeq succession, as, for instance, the oftener a halfpenny has already successively 
laving been fallen head, the less is the probability that it will fall head in the next trial, 
008; much It is rather singular that he did not from the first observe, that the extension 
real meaning of this principle to its utmost limit, namely, to the case in which the half- 
ily in view, penny should always have fallen head, would oblige us, according to his own 
that no per. rule, to class the arrival of tail among the things physically impossible, 
dition of the which never have happened, and which therefore we have no reason to be- 
mounts to ng lieve ever will happen ; and yet, according to his present argument, it will 
2 nature for be precisely in this case that tail will be most likely to happen in the next 
ell remarked trial. A sounder principle might have suggested to him, that so far as our 
ake the con- judgment is determined solely by the supposed repetition, we should be dis- 
s the events posed, on that very account, to expect rather the recurrence of head, the 
woreet took oftener it has already appeared, because that very inequality would seem to 
servation he point out an inequality in the sides favouring that event. The real cause 
appen,” says why this effect. in ordinary cases is not produced, is that we tacitly refer to, 
» the chance and are influenced by, the great number of times in which head and tail have 
followed each other indiscriminately, as well as to all the other reasons we 
¢ De. because have for believing the two sides similarly circumstanced, and the probability 
arising from this of the perfect indifference of the sides is far from being out 
woh to epat weighed by the results of a few sequences. Another error, not less extraor- 
Be sum vel dinary, was made by the same celebrated writer in the consideration of 
mn repeated experiments. If a player undertook with a halfpenny to throw head 
ra Bis ¢On* in two trials, D’Alembert observed that there were but three possible cases: 
ok vas head in the first trial; tail in the first, and head in the second ; tail in both. He, 
im and 2 therefore, asserted that the chance in favour of the player should be taken at 2, 
pi and not £, according to the ordinary rule, in which the combination of head 
ate thrown twice is taken into the account, * because as soon as head is thrown the 
be first time, it is as useless as ridiculous to throw the piece again; for the result 
ilo of the second throw has no effect upon the game, and is as foreign to it as if} 
if Wo instead of throwing the piece again, the players had gone to Rome.” D’Alemn- 
oe Bo bert’s mistake lies in supposing that his opponents were not as much aware 
mitied 10 of this as himself. Ttis true that in this game. there would be only three 
2m 3 = possible cases, but they would not be similarly circumstanced, which it is 
5 tL necessary that they should be, before an enumeration of them can furnish us 
# Qpuscules Mathématiques, vol, ii, 
A