o ON PROBABILITY. 
throw is 1; or, in common language, we should say of the first of these 
events that the odds are 2 : 1 against it, and of the second 5 : 1. 
Generally, if m =} n is the whole number of cases; and if 
m is the number of cases which are favourable to the event P, m:n are the odds 
7p RCE SR os. unfavourable... ..... 
in favour of the event P, and the probability of the event P is it 
m-n 
5. Simpson has defined the probability of an event to be the ratio of the 
chances by which the event in question may happen to all the chances by 
which it may happen or fail. In this definition the word chance must be 
understood a way of happening ; we, however, frequently say, « I left such 
a thing to chance,” or, “such a thing is entirely chance ;” these expressions, 
which are in some measure sanctioned by common use, are intended to 
signify that we are ignorant of the canses which produce the event in 
question, or that we do not influence its occurrence. 
6. When there are many events, such that one must, and only one can 
happen on any given trial, we shall call them conflicting events; and it is 
evident, from the definition of probability, that the probability that either of 
two conflicting events will happen on any single trial is equal to the sum of 
their respective probabilities; it is also evident that the sum of the proba- 
bilities of all the conflicting events which can happen on any single trial is 
expressed by unity; for, by the supposition, one of them must happen. 
Our { belief Fs founded upon the probability of the event under 
expectation 
consideration. It will often happen that our judgment is influenced by 
circumstances of too complicated or delicate a nature to be submitted to 
numerical calculation; and the conclusions with which this science furnishes 
us are true only within the limits of the errors which arise from neglecting 
these considerations. We experience the same difficulty in applying to 
physical phenomena the theories deduced from abstract principles of measure 
and motion. When all the cases which are possible, are favourable, the 
event is certain and belief becomes certitude, or knowledge. Certainty, 
which is the greatest probability, is therefore represented by unity; it must 
be distingnished from the highest degree of belief which we have called 
ceriitude ; they are often confounded with each other, while they differ in the 
same manner as probability and belief differ. In fact, we want two words 
for every stage as much as for unity, the one to express the ratio of the 
favourable to all the cases possible, the other denoting the opinion conse- 
quent on the perception of that ratio. 
7. If a bag contain no white and ten black balls, the probability of draw- 
ing a white ball is -%, or, zero; if, on the other hand, the bag contain 10 
white and no black balls, the probability of drawing a white ball is 13, or 
unity, and whatever be the number of black balls, the probability of drawing 
a white ball must be some fraction between 0 and 1, which are its limits. 
When the fraction which expresses the probability of an event is little 
different from unity, we say the event is very probable, or nearly certain ; 
when it is but little greater than 7, we say it is probable; when 4, doubtful ; 
when rather less than §, improbable ; when much less than 1, very impro- 
bable ; and when zero, impossible. : 
8. We habitually assent to propositions which have in their favour a 
probability less than unity: this degree of probability is vulgarly called 
moral certainty, an expression which is at variance with every analogy of 
language. The state of mind of a man who is aware of unfavourable events