Question : Bookie's profit

Hi Experts,

I would like to remove the element of profit from a bookie's odds to get to the 'real' odds of an event's outcome.

For example:
If a bookie gives me odds of,
PlayerA 1.8
PlayerB 4.5
Draw 3.0

This adds up to a total of (100 / 1.8) + (100 / 4.5) + (100 / 3.0) = 111%

With 11% being the bookie's profit, assuming the bookie had got the odds correct and the same amount of money had been paid out what ever the outcome.

To get back to 100% odds I had assumed you can just subtract 11% from each of the prices and this would give me 100% (true) odds.

Calculating true odds in this fashion would mean the largest percentage profit was added to PlayerA's odds and the smallest to playerB's odds. But this is not how I want the profit removed.

I want the profit removed as an inverse of the amount of money taken by the bookie on each outcome. How would I go about calculating this?

So if the bookie is to pay out 1000 points regardless of the outcome then he took:
1000 / (1.8 + 11%) on PlayerA
1000 / (4.5 + 11%) on PlayerB
1000 / (3.0 + 11%) on The draw

So the smallest percentage would be added to playerA and the largest to playerB.

Answer : Bookie's profit

Given the three bdo, you computed the bookies total probability = 111.1111...% (no roundoff this time)
It turns out that assuming that the bookie marks up each true probability by the same percentage, then that percentage is just 111.1111...% - 100% = 11.1111...%
=========================================================
Here's the reason:

Let x be bookie % multiplier on true probability (same for all events)
Let f = x/100 (the fractional representation of probability)

Find: f - given that this markup is same for all three events
Given: sum 3 bookie probabilities = 111.1111...% (or alternatively, given the three bdo values)

Recall:
P(bookie PlayerA wins) = ( 1 + f ) * P(PlayerA wins) = 100 / 1.8
        I'll call (1+f) "the bookie factor"
Solve for true probability: P(PlayerA wins) = ( 100 / 1.8 ) / ( 1 + f )
-------------------------------------------
P(PlayerA wins) = (100/1.8) / ( 1 + f )
P(PlayerB wins) = (100/4.5) / ( 1 + f )
P( Draw )           = (100/3.0) / ( 1 + f )
--------------------------------------------
Add up left hand sides (LHS) and right hand sides (RHS)
LHS = sum of 3 true probabilities = 100 %
RHS = sum of bookie probabilities / "the bookie factor"
    = ( (100/1.8) + (100/4.5) + (100/3.0) ) / ( 1 + f )
    = 111.1111... / (1+f)
--------------------------------------------
Since LHS = RHS, then
100 = 111.1111... / (1+f)
100 * (1+f) = 111.1111...
(1+f) = 111.1111... / 100 = 1.1111...
f = 1.1111... - 1 = 0.1111...
x = 100 * f = 11.1111...%
-----------------------------------------
So, x = 11.1111...% is the percent representation markup of "the bookie factor"
(Or you can just subtract 100 from 111.11... % to get  "the bookie factor" a little more quickly.