ON THE RISKS OF LOSS OR GAIN. 105 



Table I. is the probability that the resulting gain or loss 

 shall lie between M -f v and M v pounds. 



If it be absolutely known that the chances are as a 

 to b for A against B, then, as in page 81, the square root 

 ' is that of the product of 8, n } a, and b, divided by a -j- b. 

 But if all that is known be that in a -f- b previous spe- 

 lations, a gave A, and b gave B, then, as in page 92, 

 the square root is that of the product 8, n, n -J- 1, a, 

 and b } divided by a -t- 6. 



EXAMPLE. It has been observed, that of 100 specu- 

 lations, 70 yielded a profit of 20 each, and the re- 

 mainder a loss of 25 each. What is the probability 

 that in 1 50 more such speculations the total result shall 

 not differ by more than 100 from its most probable 

 amount ? 



a = 70, b = 30, n = 11, g ^ 20 gained, h = 25 lost, 

 v = 100. 



g times d = 1400 gain"! -, rt ^ . 



h times b = 750 toss j ? = <H general average of gam. 



650 gain x l = 975 probable total. 



' n llfiQ _ 0- 



- J 



_ -. 9'9999 v 9 J. 1 



h = 25 loss J 45 - J ^^ , J m* * J + 



Add 45. 

 8 x n x n +Ixax6 = 8xlx2lx70x 30 = 63,000. 



6 -^- = 63 > Visa = 25-JOO,^! 4 = -217 = t 

 Table L, if t = -217, H = -241. 



Hence it is more than 3 to 1 agaiijst the result lying 

 within the given limits. 



PROBLEM. All things remaining as in the last pro- 

 blem, what is the amount of departure from the pro- 

 bable total for being within which there is the given 

 odds p to q 9 



RULE. Turn p divided by p -f- q into a decimal 

 fraction, and find it in the column H of Table I., 

 taking out the corresponding value of t. Multiply t by 



