My maths below is wrong. thats what I get for doing it so late.

Sorry very tired while trying to type this out. Below is the correct maths for determining it. I had the divide the wrong way around.

It should read:

Lets use my $4 priced horse first. Lets say it didn't win but a horse priced $10 wins. We can then determine our horses chance of running second. We know it didn't win the race and a 10% chance did so that means we are 0.2/0.90 = 22.2% chance of running second. A horse paying $20 (5%) runs in second. We now have a 0.2/0.85= 23.5% chance of running third.

Lets say a $5 horse wins (1/5 = 20%) leaving us a 0.2/0.8 = 25 % chance of running second and lets assume a $6 chance actually ran 2nd. this leaves us with a 0.2/(0.8-.16)= 31% chance of running third.

We can do the above again with our $2.50 chance. If the horse which ran first was a $10 horse and the horse which ran 2nd was a $20 horse then our $2.50 horse has a 0.4/0.85 = 47% chance of running third. If the first horse was a $5 horse and the 2nd horse was a $6 horse then we had a 0.4/(0.8-0.16) = 62.5% chance of running third. In these instances its much higher then your 22%.