ON THE BISKS OF LOSS OR GAIN. 97 



run, may be expected to be the average produce or loss 

 per game ? 



RULE. Multiply each' gain or loss by the probability 

 of the event on which it depends ; compare the total 

 result of the gains with that of the losses: the balance 

 is the average required, and is known by the name of 

 the mathematical expectation. When the balance is 

 nothing, then the play is equal. 



For example, a person plays against a bank in the fol- 

 lowing manner : Equal stakes of one shilling each are 

 laid down, and if head be thrown twice successively he 

 wins, or if tails be thrown twice he loses. But if 

 head and tail be thrown, then another throw is to be 

 made, by which, if it be head, the player only recovers 

 his stake, but wins nothing: while, if it be tail, he 

 loses his stake. In such problems, the stake is a mere 

 evidence of solvency, with which the mathematical 

 question has nothing to do. Let us suppose, then, that 

 both parties keep their money until it is called for by the 

 result. The events on which the player gains or loses 

 are as follows (H stands for head, T for tail) : 



H H gains him a shilling. 



TT, HTT, THT lose him a shilling. 



HTH and THH give no result. 



The chance of HH is 1, and the same of TT; while 

 for each of the set HTT, THT, HTH, and THH, 

 the chance is -*. Hence the value of the prospect of 

 gain is J of one shilling ; that of loss \ -j- -J + \> or 2 f 

 a shilling : consequently the balance against the player 

 is ^ of a shilling each time, and this he would certainly 

 lose in the long run ; that is to say, the number of times 

 he loses would eventually so far exceed the number of 

 times he wins as to make the playjcost him threepence 

 per game. 



Every risk of loss must then be compensated by an 

 equal chance of gain, when the play is equal. But it 

 does not follow that equal play means prudent play ; 



