APPENDIX 693 



population, their average stature differs materially from that already found. 

 We are not at all surprised if the averages are substantially equal. There 

 are, no doubt, many causes which influence the growth of each individual 

 differently, but when they are all taken together these small disturbances 

 tend to counterbalance each other. In short, it is regularity in large num- 

 bers which we expect. While it may be common sense to expect this, we 

 shall later give a mathematical measure known as the " probable error" to 

 indicate what deviations we should expect in results such as averages derived 

 from a random sample. This discussion leads us to the following definition 

 of probability. 



Definition. If, in the long run, out of n possible cases in each of which 

 an event occurs or fails to occur, it occurs n\ times and fails to occur n n^ 

 times, the probability that the event occurs on a given occasion in question 



is , and the probability that it fails to occur on a given occasion is 



n n 



In framing this definition we idealize our actual experience. We say 

 the probability of a penny's turning up heads is one half. This may be 

 looked upon as an answer to the following question : What proportion of 

 the pennies tossed should we expect to find with heads turned up if we 

 should continue tossing indefinitely ? 



This idealization, for purposes of definition, is analogous to the idealiza- 

 tion of the crude chalk mark into the straight line of geometry. Since the 



sum of the probabilities of occurrence and failure is H = I, the 



n n 



number i is the symbol of certainty. The expression " relative frequency " 

 conveys fairly well the idea of probability. 



The following corollary is often easier to apply than the definition. 



Corollary. If the entire number of possible cases in which an event is in 

 question can be analyzed into n' cases, each of which is equally likely, and 



m' is the number of these cases in which the event occurs, then is the 

 probability of the event. 



Thus, in tossing two pennies, what is the probability that one will be 

 heads and one tails ? 



There are four different ways in which the pennies may fall : Head and 

 tail, tail and head, head and head, tail and tail. Two of these ways lead 

 to the occurrence of the event. Hence f = \ is the desired probability of 

 one head and one tail. 



Combination of probabilities. The probability that all of a set of independ- 

 ent events will occur on any occasion in which all of them are in question is 

 the product of the probabilities of the single events. 



Proof. Let/p/.,, . . ., p r be the separate probabilities of r events. Out 

 of a great number, n, of cases, the first will happen on p^n occasions. Out 

 of these the second will happen on p 2 (p^ri) occasions. Continuing this 

 process, and applying the definition of probability, the theorem is at once 

 established. To illustrate, suppose that among a population of a hundred 



