90 ESSAY ON PROBABILITIES. 



gives A a greater facility in the second than in the first, 

 or a greater probability of being drawn at any one trial ? 

 Or, if in a first set of a -f b trials, A happen a times 

 and B happen b times ; and if in a second set of a' -f b' 

 trials,, A happen a times, and B happen &' times ; and if 

 a' be a larger proportion of a x -f b' than a is of a 4- b ; 

 required the presumption that there was a greater chance 

 of drawing A at a single trial in the second set than in 

 the first ? 



RULE. Divide the cube of the sum of a and b by 

 twice their product : do the same with a' and 6 X : mul- 

 tiply the two results together, and add them together : 

 divide the product by the sum, and extract the square 

 root of the quotient. 



Divide a by of + &', and a by a -f b, and subtract 

 the less result from the greater. Multiply the difference 

 by the square root previously found, and let the product 

 be t. Then the H corresponding to t, increased by 1, 

 and divided by 2, is the presumption required. 



In the example a = 50, b = 30, a = 1 12, &' = 6l. 



80 x 80 x 80 = 512000 2 X 50 X 30 = 3000, 51200 = 1707 



3000 



173 x 173 X 173 = 5177717 2 X 112 X 61 = 13664 j 5177717 = 378-9 



13664 

 1707 X 378-9 = 64678-23, 170'7 + 378*9 =549 '6 



64678-23 = 117.7, v ? H77 = 10-85 

 549-6 

 tyjfjj= -6474 -6250 = 0224 



0224 X 10-85 = -24304 = t, H = -2689 



J (H -f- 1) = '635, the probability required ; and it is 

 therefore about 16 to 9 in favour of the excess of As 

 at the second set of trials not being accidental fluctua- 

 tion, but arising from some new circumstance or dif- 

 ferent arrangement of the old ones. 



If, in 1000 trials, A should happen 520 times, and 

 B 480 times, there is strong presumption that in any fu- 

 ture number of trials the whole number will be divided 

 among As and Bs nearly in the proportion of 520 to 



