232 THE PRINCIPLES OF SCIENCE. 



We must distinguish comparative from absolute pro- 

 babilities. In drawing a card casually from a pack, there 

 is no reason to expect any one card more than a.ny other. 

 Now, there are four kings and four queens in a pack, so that 

 there are just as many ways of drawing one as the other, 

 and the probabilities are equal. But there are thirteen 

 diamonds, so that the probability of a king is to that of 

 a diamond as four to thirteen. Thus the probabilities 

 of each are proportional to their respective numbers of 

 ways of happening. Now, I can draw a king in four 

 ways, and not draw one in forty-eight, so that the pro- 

 babilities are in this proportion, or, as is commonly said, 

 the odds against drawing a king are forty-eight to four. 

 The odds are seven to seventeen in favour, or seventeen 

 to seven against the letters R,o,m,a, accidentally forming 

 a significant word. The odds are five to three against 

 two tails appearing in three throws of a penny. Con- 

 versely, when the odds of an event are given, and the 

 probability is required, take the number in favour of the 

 event for numerator, and the sum of the numbers for 

 denominator. 



It is obvious that an event is certain when all the 

 combinations of causes which can take place produce 

 that event. Now, if we were to represent the pro- 

 bability of any such event according to our rule, it would 

 give the ratio of some number to itself, or unity. An 

 event is certain not to happen when no possible combina- 

 tion 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 i, 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 compound event, we mean an event which may be 



