230 THE PRINCIPLES OF SCIENCE. 



there is a margin of error which can only be safely treated 

 by the principles of probability. 



The method which we employ in the theory consists 

 in calculating the number of all the cases or events 

 concerning which our knowledge is equal. If we have 

 even the slightest reason for suspecting that one event 

 is more likely to occur than another, we should take this 

 knowledge into account. This being done, we must 

 determine the whole number of events which are, so far 

 as we know, equally likely. Thus, if we have no reason 

 for supposing that a penny will fall more often one way 

 than another, there are two cases, head and tail, equally 

 likely. But if from trial or otherwise we know, or think 

 we know, that of 100 throws 55 will give tail, then the 

 probability is measured by the ratio of 55 to 100. 



The mathematical formulae of the theory are exactly the 

 same as those of the theory of combinations. In this 

 latter theory, we determine in how many ways events may 

 be joined together, and we now proceed to use this know- 

 ledge in calculating the number of ways in which a certain 

 event may come about, and thus defining its probability. 

 If we throw three pennies into the air, what is the proba- 

 bility that two of them will fall tail uppermost "? This 

 amounts to asking in how many possible ways can we 

 select two tails out of three, compared with the whole 

 number of ways in which the coins can be placed. Now, 

 the fourth line of the Arithmetical Triangle (p. 208) gives 

 us the answer. The whole number of ways in which we 

 can select or leave three things is eight, and the possible 

 combinations of two things at a time is three ; hence the 

 probability of two tails is the ratio of three to eight. 

 From the numbers in the triangle we may draw all the 

 following probabilities : 



One combination gives o tail. Probability ^. 

 Three combinations give i tail. Probability f . 



