204 THE PRINCIPLES OF SCIENCE. [chap. 



according to our rule, it gives the ratio of some nunilier to 

 itself, or unity. An event is certain not to happen when 

 no possible combination of causes gives the event, and the 

 ratio by the same rule becomes that of o to some number. 

 Hence it follows that in the theory of probability certainty 

 is expressed by l, and impossibility by o ; but no mystical 

 meaning should be attached to these symbols, as they 

 merely express the fact that all or no possible combinations 

 give the event. 



By a comioound event, we mean an event which may be 

 decomposed into two or more simpler events. Thus the 

 firing of a gun may be decomposed into pulling the 

 trigger, the fall of the hammer, the explosion of the 

 cap, &c. In this example tlie simple events are not 

 independent, because if the trigger is pulled, the other 

 events will under proper conditions necessarily follow, and 

 their probabilities are therefore the same as that of the 

 first event. Events are independent when the happening 

 of one does not render the other either more or less 

 probable than before. Thus the death of a person is 

 neitlier more nor less probable because the planet Mars 

 happens to be visible. When the component events are 

 independent, a simple rule can be given for calculating 

 the probability of the compound eveut, thus — Midtiply 

 together the fractions expressing the prohahilUies of the 

 independent componeid events. 



The probability of throwing tail twice with a penny is 

 ^ X 5, or \; the probability of throwing it three times 

 running is ^ x |- x ^, or ^ ; a result agreeing with that 

 obtained in an apparently different manner (p. 202). In 

 fact, when we multiply together the denominators, we 

 get the whole number of ways of happening of the com- 

 pound event, and when we multiply the numerators, we 

 get the number of ways favourable to the required event. 



Probabilities may be added to or subtracted from each 

 other under the important condition that the events in 

 question are exclusive of each other, so that not more than 

 one of them can ha.ppen. It might be argued that, since 

 the probability of throwing head at the first trial is \, and 

 at the second trial also \, the probability of tlirowing it 

 in the first two throws is ^ + \, or certainty. Not only is 

 this result evident] v absurd, but a repetition of the process 



