202 THE PRINCIPLES OF SCIENCE. [CHAP. 



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. It is the comparative numbers of 

 ways in which events can happen which measure their 

 comparative probabilities. If we throw three pennies 

 into the air, what is the probability 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 Arith- 

 metical Triangle (p. 184) 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 similarly draw all the following probabilities : 



One combination gives o tail. Probability . 



Three combinations gives I taiL Probability f . 



Three combinations give 2. tails. Probability f . 



One combination gives 3 tails. Probability . 

 We can apply the same considerations to the imaginary 

 causes of the difference of stature, the combinations of 

 which were shown in p. 188. There are altogether 128 

 ways in which seven causes can be present or absent. 

 Now, twenty-one of these combinations give an addition 

 of two inches, so that the probability of a person under 

 the circumstances being five feet two inches is ^. The 

 probability of five feet three inches is ^T S ; of five feet 

 one inch -j-J^; of five feet y^, and so on. Thus the 

 eighth line of the Arithmetical Triangle gives all the 

 probabilities arising out of the combinations of seven causes. 



