1> ESSAY ON PROBABILITIES. 



sets in which head should not appear till the tenth throw, 

 and one in which no such thing should take place till 

 the eleventh, and if we calculate the total amount 

 which would have been realised had the average case 

 occurred, we shall find it to be 1 1 per game. In the 

 experiment in question, it would have produced < 10. In 

 precisely the same way, sets each consisting of 2" games 

 would have realised n per game (2048 is the eleventh 

 power of 2). 



I now come to the estimation of the chances of 

 fluctuation in loss or gain, meaning by fluctuation 

 any departure from that general average to which the 

 results of more and more trials will continually approach. 

 It has been assumed in what precedes, that the propor- 

 tions which the fluctuation will bear to the whole will 

 diminish without limit as the number of speculations 

 increase. The following problems are easy deductions 

 from those in the last chapter. 



PROBLEM. It is known to be a to b for A against B. 

 A is an event which brings a loss or gain of g pounds ; 

 B is another event which brings a loss or gain of h 

 pounds. What is the general average of such trials ; 

 and what is the chance that in n times a -f b trials, the 

 result as to loss or gain shall differ from the general 

 average by not more than v pounds. 



RULE. Find g times a, and h times 6, and if g and 

 h be both gains or both losses, take their sum ; but it 

 one be a gain and the other a loss, take the difference, 

 counting it gain or loss, according as the term which 

 contained the gain or the loss was the greater. Multiply 

 the result by n, wjiich gives the most probable total result 

 (call this M). The general average is the n ( a -f- 6)th 

 part of this ; or, more simply, the (a + &)th part of 

 the balance of g times a and h times b. Take the differ- 

 ence of g and h, if of the same name, or their sum, it 

 of opposite names, and by it divide v. Take one more 

 than twice the quotient. Having found this result, 

 divide it by a square root immediately to be described, 

 and let the quotient be t. Then the value of H in 



