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Assure A Sure Bet...You Sure?

A gambler, dissatisfied with his tendency to lose on bets, decides to be more cautious in his betting on a pending boxing match. The stated odds are 5 to 1 in favor of the champion.

Deciding he will place money on both the champion and the challenger, the gambler needs to determine the smallest sum of money that he need wager on each in order to insure himself a return of at least $10, no matter who wins the bout.

With this restriction in mind, what amounts should the gambler bet on each man?

Note: The problem as stated has considerable ambiguity in it. That is, does the gambler want to win $10, regardless of which fighter wins...or does the gambler want to walk away with a winnings of $10, taking into account the amount bet on the non-winner as well?

Try working the problem under both assumptions...and perhaps a surprise lies ahead!

 

Source: Adapted from A. Douglis' Ideas in Mathematics, 1970, p. 311.


Hint: Be sure you understand the role of odds here...for example, the idea of "odds against" is a good way to understand what winnings will occur.

If the odds are 1 to 5 against the challenger, then if the challenger wins, you will win (5/1)x dollars (where you bet x dollars on the "winning" challenger) and will win (1/5)y dollars (where you bet y dollars on the "winning" champion).

Though the original bet was x+y dollars, only the bet (x or y) on the winning boxer is returned along with your other winnings.

Finally, linear inequalities should be helpful in exploring this problem.

 


Solution Commentary: I will let you argue this one out. The text source of this problem argues that the bet should be $52, or $50 on the champion and $2 on the challenger. Does this make sense? What assumptions arte being made?