Suppose four horses - A,B,C, and D - are entered in a race and the...

Logan Wyatt

Logan Wyatt

Answered

2022-07-10

Suppose four horses - A,B,C, and D - are entered in a race and the odds on them, respectively, are 5 to 1, 4 to 1, 3 to 1, and 2 to 1. If you bet $1 on A, then you receive $6 if A wins, or you realize a net gain of $5. You lose your dollar if A loses. How should you bet your money to guarantee that you can always make money no matter which horse wins?

Answer & Explanation

trantegisis

trantegisis

Expert

2022-07-11Added 20 answers

Each of the horses has a certain return when they win, A is a 6 × return, B is 5 ×, C 4 ×, and D 3 ×. In general say you have a set of horses H i that each have a return r i , and you bet some total amount B with B i on each horse H i . As long as B i B r i for all i, you will not lose money. That is because one of the H i will win, and that bet will give you B i r i B, so you end up with at least as much as you started. To maximize your guaranteed winnings, you want to maximize the minimum B i r i , as B i r i B is your net profit. This will happen when each B i r i is equal, which occurs when
B i = B r i 1 r i
and your guaranteed profit is
B 1 r i B
This is why in a real horse race 1 r i will always be greater than 1(if for some reason it's not go make a big bet!). In this question it was only .95 allowing you to make a nice little profit!

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