ON DIRECT PROBABILITIES. 49 



Principle II. The probability of any number of in- 

 dependent events all happening together, is the product 

 of their several probabilities. 



Principle III. The probability of two events arriving 

 together being known ; and also,, that of one of them : the 

 probability of the other is found by dividing the first 

 mentioned probability by the second. 



Principle IV. When an event may happen in several 

 ways, whether equally probable or not, the probability 

 of the event is the sum of the probabilities of its hap- 

 pening in the several different ways. 



The best way of illustrating the last principle is by 

 beginning with one of the numerous errors into which 

 we may fall, either in proceeding towards it, or applying 

 it. Let there be an event which may happen in two 

 different ways, for each of which there is an even chance. 

 Then, according to the principle, it is certain ( J- -f -J- = 1 ) 

 that the event must happen. In this there is nothing 

 inconsistent. Let there be a lottery containing ten white 

 balls, and let them be sub-divided into two sets of five 

 by a mark. Then there are two ways of drawing a 

 white ball, for each of which there is an even chance ; 

 namely, I may choose a ball of one mark, or of the other. 

 But it is evidently certain that in this case a white ball 

 must be drawn. Now, suppose an event can happen in 

 three different ways, for each of which there is an even 

 chance. Then the event k more than certain (-J- + A 

 H- -J- = 1 1) ; which is absurd. But the absurdity is in 

 the supposition : an event can only have an even chance 

 of happening in one particular way, when that way 

 involves half of the total number of individual cases of 

 the event ; and it is impossible that three different ways 

 of arriving can each contain half of the whole number 

 of possible cases. Consequently, when we have made a 

 calculation of the probabilities which different ways of 

 arriving give to an event, there has certainly been an 

 error if the sum of the probabilities exceed unity. 



But when we throw three half -pence into the air, are 

 there not three different ways of throwing head, for each 



