I am trying to learn about expected value calculations in sports betting... As an example, say I am a sports bettor, and there is a race with 3 horses, $a$, $b$ and $c$ and I know their respective true probabilities, $\Pr(a)$, $\Pr(b)$ and $\Pr(c)$. Assume that a bookkeeper is offering the implied probabilities (consider these the probabilities decided by the market), $\hat{\Pr(a)}$, $\hat{\Pr(b)}$ and $\hat{\Pr(c)}$. I am trying to find the maximum possible expected value, $EV_{max}$, by deciding the wagers, $w_a$, $w_b$ and $w_c$ that I should place on the respective horses. I framed this as a maximisation function (essentially return weighted by probability of event), but am unsure how to select the optimal wagers.
$EV_{max}=\\max(\Pr(a)(\frac{w_a}{\hat{\Pr(a)}}-w_a-w_b-w_c)+\\\Pr(b)(\frac{w_b}{\hat{Pr(b)}}-w_a-w_b-w_c)+\\\Pr(c)(\frac{w_c}{\hat{Pr(c)}}-w_a-w_b-w_c))$
First, this is not betting advice. Second, you should have some limit on the total amount bet, or you may be able to increase your expected winnings simply by betting more. Let W be the total amount you will wager. $w_a$ the wager on horse $a$, $w_b$ the wager on horse $b$, and I a simplifying this to a two-horse race. Let $r_a$ be the return for betting one dollar on horse $a$. If you think odds, if the odds are 2-1, betting 1 wins you 2, 3:2, betting 1 gets you 1.5. Let $p_a$ be the probability of horse $a$ winning.Then your expected winnings are $$p_a(w_ar_a)-(1-p_a)w_a +p_b(w_br_b)-(1-p_b)w_b.$$ $$w_a[p_a(r_a+1)-1]+w_b[p_a(r_a+1)-1].$$ So the linearity shows that which ever bet has the higher expected return, put all your money on that. Of course we can simplify this by noting that $w_b=W-w_a$ and $p_b=1-p_a.$