ON DIRECT PROBABILITIES. 51 



1st event ; certainly happens,, and gives either H or T. 

 Probability of H, |. 



2nd event ; does not certainly happen, but is con- 

 tingent upon the first throw being T ; and gives either 

 H or T. Probability \ that the throw is made, that 

 if it be made it gives H ; probability that the throw is 

 made, and that, being made, it gives H, -J- x -J- -J. 



3rd event; contingent upon the two first throws 

 giving T and T, of which the chance is | . Probability 

 of winning at this throw, | X -J- = -g. 



4th event ; contingent upon the three first throws 

 giving T, T, T. Probability of winning at this throw, 



4 * i = /* 



It is obvious enough, when stated, that every con- 

 tingency must enter into the consideration of a question ; 

 so that if some of the circumstances depend upon the 

 manner in which preceding contingencies arrive, this 

 circumstance itself influences the method of proceeding. 

 If we wish to avoid the necessity of considering a con- 

 tingency the trial of which is itself contingent, that is, if 

 we wish to make a contingency certain, we must in- 

 troduce all the new events which such change of con- 

 tingency into certainty brings with it. The whole 

 problem is exactly the same as if we made the four 

 throws certain, and made the gain dependent upon one 

 head or more being thrown : but we revert again to the 

 former state of the question, if we agree to mark the first 

 H as the winner. In this point of view, the distinction 

 between the two is evidently iin material. 



EXAMPLE. There are seven lotteries, as follows (W 

 means white, B black) : 



L(2W ' 3B) 



and the conditions of drawing are the following. I. i's 

 drawn, and then II. or III., according as I. gives W or B. 

 If II. be drawn, then IV. or V. is to be drawn, according 

 as II. gives W or B. But if III. be drawn, then VI. or 



E 2 



