94 ESSAY ON PROBABILITIES. 



for the prospect, one time with another. But ^ is the 

 probability of throwing the ace. 



PRINCIPLE. Multiply the sum to be gained by the 

 fraction which expresses the chance of gaining it, and 

 the result is the greatest sum which should be given for 

 the chance. 



PROBLEM. I am to gain by the throw of a die as 

 many pounds as there are spots on the face which is 

 thrown, with this exception, that I am to lose as many 

 pounds as there are spots on both the face in question 

 and on that of another die which is thrown at the same 

 time, if the two give doublets. What should I give 

 for the chance in question ? 



Call these dice P and Q : then the chance of an ace 

 from P, and something not an ace from Q, is the pro- 

 duct of i and |> or ^ ; and the same for any given 

 throw from P, and not the same from Q. Consequently, 

 the chance of winning, independently of the chance of 

 losing, is worth 



3 5 g x 1 + 3 5 5 x 2 + g * g x 3 + 3 3 g x 4 + 3 5 g x 5 + 3 5 g x 6, 



or - 3 5 6 - x 21, or 2-|-J. But if the chance of winning 

 must be paid for, the chance of losing must be paid. 

 Now the chance of throwing any one pair of doublets 

 is the product of J and ^, or ^ : whence the sum I 

 must receive if I pay 2^ (or not pay out of the 2-\-$, 

 which is the same thing) is 



3S X 2 + 35 x 4 + 38 x 6 + & X 8 + J a X 10 + 3 'g X 12, 



or 1 J. Consequently, I must pay only If. If I were 

 to stand a million of such hazards, paying l-| for each, 

 I might expect my ultimate gain or loss to be extremely 

 small, compared with the whole sum risked. If I had 

 besides a very small profit upon each, I might be sure 

 of winning. 



We have in the last chapter considered the amount of 

 fluctuation for which there is a given probability : we 

 will now look at the percentage of fluctuation, that i^, 



