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THE TURF 193 



Upon the same two events, what odds 

 may B bet A that the latter does not lose 

 both ? Set down for the former J, for the 

 latter T 4 T : 



Then \ x T * T = ^r 5 an <* 22 - 4= 18 : 



therefore 1 8 to 4 = 9 to 2 is the odds. 



Proof by Hedging. B bets first the sum 

 to which he has laid his odds, namely, two 

 pounds, which he wins ; and then taking 

 7 to 4 on the second event, he wins 2 + 7 = 9, 

 which pays the nine pounds he lost to A ; 

 and had more favourable odds been offered, 

 B must have been a winner without risk of 

 losing. 



When three distinct events are pending, 

 on the first of which the betting is even ; 

 on the second 3 to 2 in favour of A, and 

 the third 5 to 4 ; what odds should B 

 lay A that the latter does not name all 

 the winners ? The first is expressed by J, 

 the second by f , and the third by : 



Therefore, 



4 x f if (by cancelling) \ ; and 6- 1 = 5: 



hence the odds are 5 to i. 



Proof by Hedging. B begins to hedge by 

 betting an even two pounds that A wins the 



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