THE THEORY OF PROBABILITY. 2S3 



distinguished into two or more simpler events. Thus 

 the firing of a gun may be distinguished into pulling the 

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

 &c. In this example the simple events are not' inde- 

 pendent, 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 neither more 

 nor ]ess 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 event, thus Multiply together the frac- 

 tions expressing the probabilities of the independent 

 component events. 



The probability of throwing tail twice with a penny 

 ^ s i x i> or 4 > the probability of throwing it three times 

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

 obtained in an apparently different manner (p. 230). In 

 fact when we multiply together the denominators, we get 

 the whole number of ways of happening of the compound 

 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 happen. It might be argued that as 

 the probability of throwing head at the first trial is -Jr, and 

 at the second trial also ^, the probability of throwing 

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

 is this result evidently absurd, but a repetition of the 

 process would lead us to a probability of i^ or of any 

 greater number, results which could have no meaning 

 whatever. The probability we wish to calculate is that of 



