x.] THE THEOEY OF PROBABILITY. 203 



Rules for the Calculation of Probabilities. 



I will now explain as simply as possible the rules 

 for calculating probabilities. The principal rule is as 

 follows : 



Calculate the number of events which may happen 

 independently of each other, and which, as far as is 

 known, are equally probable. Make this number the 

 denominator of a fraction, and take for the numerator 

 the number of such events as imply or constitute the 

 happening of the event, whose probability is required. 



Thus, if the letters of the word Roma be thrown down 

 casually in a row, what is the probability that they will 

 form a significant Latin word ? The possible arrange- 

 ments of four letters are 4 x 3 x 2 x I, or 24 in number 

 (p. 178), and if all the arrangements be examined, seven 

 of these will be found to have meaning, namely Roma, 

 ramo, oram, mora, maro, armo, and amor. Hence the 

 probability of a significant result is -/ T . 



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 any 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 probabili- 

 ties of each are proportional to their respective numbers 

 of ways of happening. Again, 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 E,o,m,a, accidentally forming a 

 significant word. The odds are five to three against two 

 tails appearing in three throws of a penny. Conversely, 

 when the odds of an event are given, and the probability is 

 required, take the odds in favour of the event for numerator, 

 and the sum of the odds for denominator. 



It is obvious that an event is certain when all the com- 

 binations of causes which can take place produce that 

 event. If we represent the probability of such event 



