Estimating implied probability based on prediction betting odds

Question

I am attempting to estimate prediction betting market efficiency for a project, and I am hoping for assistance with a couple of questions.

The prediction market makers add a commission to the betting odds offered, and I am trying to work out the true implied odds based on the odds offered.

Odds format: The odds are the payout (profit). Therefore, if I bet a \$1 on outcome A, and the odds are 1.00, I will get my \$1 back, plus the odds (\$1) for a total of $2.

$\textbf{1) Starting with a binary random variable: Pr(A) + Pr(B) = 1}$

Calculating implied Pr if the market maker takes no commission:
$E(A) = 0 = [1-Pr(A)]*1 - Pr(A)*Odds_A$
$Pr(A) = \frac{1}{Odds_A + 1}$
$Pr(B) = \frac{1}{Odds_B + 1}$
$Pr(A) + Pr(B) = \frac{1}{Odds_A + 1} + \frac{1}{Odds_B + 1} = 1$

If the market maker embeds a commission into the odds offered, then: $\frac{1}{Odds_A + 1} + \frac{1}{Odds_B + 1} - 1 = C$
Implied odds are: $Pr(A) = \frac{1}{Odds_A + 1} - C * \frac{\frac{1}{Odds_A + 1}}{\frac{1}{Odds_A + 1} + \frac{1}{Odds_B + 1}}$ and $Pr(B) = \frac{1}{Odds_B + 1} - C * \frac{\frac{1}{Odds_B + 1}}{\frac{1}{Odds_A + 1} + \frac{1}{Odds_B + 1}}$.

$\textbf{Questions}$

1. I'm struggling with what "C" represents. I know it is the market maker's commission, but more specifically, what does it represent? I'm guessing it is some form of % of payout, but I'm struggling to derive meaning from this term.

2. Can someone provide a proof as to why $\frac{1}{Odds_A + 1} - C * \frac{\frac{1}{Odds_A + 1}}{\frac{1}{Odds_A + 1} + \frac{1}{Odds_B + 1}}$ correctly allocates the correct amount of commission to Pr(A), so that the estimate of Pr(A) is the true implied Pr(A)?

$\textbf{2) Three possible outcomes: Pr(A) + Pr(B) + Pr(D) = 1}$
Calculating implied Pr if the market maker takes no commission:
$E(A) = 0 = [1-Pr(A)]*1 - Pr(A)*Odds_A$
$Pr(A) = \frac{1}{Odds_A + 1}$
$Pr(B) = \frac{1}{Odds_B + 1}$
$Pr(D) = \frac{1}{Odds_D + 1}$
$Pr(A) + Pr(B) + Pr(D) = \frac{1}{Odds_A + 1} + \frac{1}{Odds_B + 1} + \frac{1}{Odds_D + 1} = 1$

If the market maker embeds a commission into the odds, then: $\frac{1}{Odds_A + 1} + \frac{1}{Odds_B + 1} + \frac{1}{Odds_D + 1} - 1 = C$

Using the same process as in a binary outcome to calculate the implied Pr: $Pr(A) = \frac{1}{Odds_A + 1} - C * \frac{\frac{1}{Odds_A + 1}}{\frac{1}{Odds_A + 1} + \frac{1}{Odds_B + 1} + \frac{1}{Odds_D + 1}}$. Unfortunately, this equation is incorrect and doesn't output the implied Pr(A).

$\textbf{Questions}$

1. Why did this equation work for a binary outcome, but not for a random variable with three outcomes?

2. Is anyone aware of an equation for calculating the implied Pr of a R.V with more than 2 outcomes by removing the impact of a market maker's commission?