80 ESSAY ON PROBABILITIES. 



Suppose the question to be that of page 77) namely, 

 what is the probability that the number of aces in 6000 

 throws shall lie within 50, one way or the other, of the 

 probable mean 1000? Now,, 18'97 -f- '5 is 19'47, and 

 50 + -5 is 50-5, and 50-5 divided by 19'47 gives 

 2-594, which being t (in Table II.), K is '91981, ex- 

 tremely near to the result in page 77- 



There are, therefore, two distinct methods of treating 

 these problems, connected with the two tables : and this 

 is a great advantage, since it is a very strong presump- 

 tion of a correct answer, when the results of the tables 

 agree. The problems III. and IV. .being of great import- 

 ance, I shall now recapitulate their details, with the ad. 

 dition of some new phraseology. Let the instance be 

 6000 throws of a die, and the event A the arrival of an 

 ace, and B the arrival of some other face. The most 

 probable number of aces is 1000, though the arrival of 

 that exact number is not probable in the common sense 

 of the word. There will then most likely be a de- 

 parture from the number 1000 in the number of aces 

 thrown ; of which departure we are now entitled to 

 say, that it is very improbable it should be considerable. 

 Let the term neutral departure mean that degree of de- 

 parture for which it is just an even chance that the 

 actual event shall be contained within its limits : in 

 the present instance it is 18'97- We may explain the 

 fraction as follows : suppose a person to receive 100/. 

 for every unit by which the number of aces falls short 

 of or exceeds 1000. Then, supposing him to try this 

 stake a great many times, he will in the long run re- 

 ceive less than 1897/. at a trial, as often as he receives 

 more. But his receipts will oftener exceed than fall 

 short of 1800/. ; while they will oftener fall short of 

 than exceed 1900/. Roughly speaking, there is here the 

 same probability that the aces shall not lie between 

 1000-19 and 1000+ 19 (both inclusive), and that 

 they shall lie between these numbers. 



In all these problems there is a square root to be 

 found, which we call the square root, as there is no other. 



