88 ESSAY ON PROBABILITIES. 



another, namely, that it is easier to produce the first 

 than the second. In our problems,, however, the 

 facility is not that arising from art, but from previous 

 (it may be accidental) distribution of means. The 

 word expectation will lie applied throughout this work 

 to that state of things for the production of which 

 there is an even chance. If (p. 79)^ 6000 throws be 

 made with a die, it is an even chance that the number 

 of aces lies between 981 and 1019 : the odds are 

 against any smaller amount of departure on both sides 

 of the probable mean, and against any greater amount ; 

 this is then our expectation of the number of aces. 



When one of two possible events happens oftener than 

 the other, it being understood that one, and only one^ 

 can happen each time, we are led to suppose that the 

 excess of one event is the consequence of some arrange- 

 ment which would, had we known it, have made us 

 count that event more probable than the other. If A 

 or B must happen, and if in a thousand trials the As 

 outnumber the Es very much, we feel perfectly cer- 

 tain that such must have been the case. The theory 

 of probabilities confirms this impression, as will appear 

 by the solution of the following 



PROBLEM. In a-\- b trials, the number of As was 

 , and that of Bs was b. If a exceed b considerably *, 

 required the presumption that there was at the outset 

 a greater probability of drawing A than of drawing B, 

 in any one single trial ? 



RULE. Divide the difference of a and b by the 

 square root of twice their sum, and let the result be t. 

 Find (page 72) the H' corresponding to t : multiply 

 the result by the sum of a and b, and divide by the 

 product of 8, ^ and the square root of the product of 

 a and b. The result subtracted from unity gives the 

 answer required. Suppose, for instance, that out of 

 50 trials A occurs 32 times, and B 18 times. Then, 



* In order that the result may be very correct, a must exceed b so much 

 that the excess of a above b, multiplied by itself, may considerably exceed 

 the sum of a and 6. 



