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Question

Horses X,Y and Z are entered in a three-horse race in which ties are not possible. The odds against X winning are 3:1 and the odds against Y winning are 2:3, what are the odds against Z winning? (By "odds against H winning are p:q" we mean the probability of H winning the race is qp+q.)

Accepted Answer

Because the odds against X are 3:1, the chance of X losing is 34. Since the chance of X losing is the same as the chance of Y and Z winning, and since the odds against Y are 2:3, Y wins with a probability of 35. Then the chance of Z winning is 34 - 35 = 320. Therefore the odds against Z are 17:3.