Suppose there are $10$ horses in a race and a bookmaker quotes odds of $r_i$ to $1$ against horse $i$ winning. Let $p_i = \frac{1}{r_i+1}, \ i=1,\dots,10$, so each $p_i$ is between $0$ and $1$. Let the summation be $p_1 + p_2 + \cdots + p_{10}$
a) do you expect that the summation is greater than, smaller than, or equal to $1$? Why?
b) suppose the summation were less than $1$. Could you take advantage of this? How?
[hint: by betting on all $10$ horses in the race, a better can win a constant amount of money, regardless which horse win.]
I'm having some trouble understanding what the summation in this case even stands for. I know that a) is less than one by simply plugging in but since I don't understand what the summation represents, I can't really move on to b).
Any and all help is appreciated!
a- without even going into an analysis of the series, we can see that the summation will be > 1, as the ist 3 terms themselves [0.5 + 0.33 + 0.25 ] > 1
b- if the summation were less than 1, we could place equal bets on all the horses.
since the actual probabilities would be more on the whole since they have to sum up to 1, we should gain