In a simple example, if the bookmaker would rate an event as 1.91 for side-A and 1.91 for side-B, the honest odds would be 2 and 2 respectively. If he wanted to donate money in the long run and rated the event as 2.5 and 2.5, the honest odds would be 2 and 2 as well.
Here’s an example where we also calculate the honest odds with percentages: let say a bookmaker rates an event as 1.25 for side-A and 3 for side-B. 1.25 implies a probability of 80% and 3 implies 33.(3)%. We know this doesn’t reflect the true probability of those outcomes because it totals 113.(3)%. We have to raise both odds in a proportionally similar manner in order to get them to total 100%. We can’t just raise the odds for side-B to 5 (20%) and say that 1.25 and 5 would be the honest odds, because in our case they wouldn’t be. So… for the first one:
We know that a 80% probability corresponds to 113.(3)% total.
We want to know which probability (let’s call it x) corresponds to 100% total.
We’ll use cross multiplication in order to find x.
x * 113.(3) = 80 * 100
Therefore x = 70.59%
70.59% can be translated to decimal odds 1.417 (by doing 100/70.59). These are the honest odds for side-A.
Now we have to follow the same process for the second percentage:
33.(3)% corresponds to 113.(3)% total.
x corresponds to 100% total.
I’ll skip the calculation but you can do this one both ways, with percentages and directly with odds. In both cases you should arrive at honest odds of 3.4 (29.41%).