106 ESSAY ON PROBABILITIES. 



the square root above mentioned, and having subtracted 

 1, divide the remainder by 2. Multiply the quotient 

 by the difference or sum of g and h, according as they 

 are of the same or different names, and the product is 

 the answer required. 



EXAMPLE. In the preceding example, within what 

 departure from .975 is it 10 to 1 that the result shall 

 be contained? 



p= 10,?= l,p + (p + q) ='9091, -9091 x 25'1 = 22-8184. 

 (22-8184 1) = 10-9092 10-9092 x 45 = 490'9/. 



It is, then, 10 to 1 that the balance of 150 speculations 

 shall lie between 975 + 491 and 975 491 pounds, or 

 1466/. and 484/. Even such a case shows the effect of 

 multitude in diminishing risks. The possible extremes 

 of the problem (or the result of the problem itself, if 

 we supposed one speculation instead of 150) are a gain 

 of 3000/., and a loss of 37501. 



I will now add an example which will tend to show 

 the ultimate effect of gambling against a bank with a 

 slight mathematical advantage in its favour. Suppose 

 the game is such, that at each trial it is 30 to 29 * that 

 the bank shall win, the stake on both sides being one 

 sovereign. Here a -f- b is 59, and making n a whole 

 number for convenience of calculation, let n = 50, or 

 let 2950 games be tried. The bank has each time a 

 mathematical advantage (page 97) of ^ --, or 

 of a sovereign, and will, in the long run, realise 50 

 upon 2950 games. What are the chances in favour of 

 the individual fluctuation of this one set of 2950 games 

 leaving the bank without any profit, and with more or 

 less loss ? To apply the preceding rule, we must first 

 ask what are the chances that the departure from the 

 probable total of 50/. shall not exceed 507. ; that is, 

 that the bank shall realise between 01. and 100/. Here 

 we have 



* This supposition is more in favour of the player than is often the 

 case. 



