Bookmakers betting book maths question

This question relates to some coding we are doing. I would be really grateful if some of you maths experts can come up with a formula for the following:

The question is this: In a horse race you might have six runners with odds like this:

Horse A - odds of 1/2 (that is decimal odds of 1.5)
Horse B - odds of 2/1 (decimal odds of 3)
Horse C - odds of 20/1 (decimal odds of 21)
Horse D - odds of 20/1 (decimal odds of 21)
Horse E - odds of 20/1 (decimal odds of 21)
Horse F - odds of 20/1 (decimal odds of 21)

Now a bookie will see the percentage of this book:
Horse A - equates to 66.67% (that is 100/decimal odds i.e: 100/1.5 = 66.67)
Horse B - equates to 33.33% (that is 100/3 = 33.33)
Horse C - equates to 4.76% (that is 100/21 = 4.76)
Horse D - equates to 4.76%
Horse E - equates to 4.76%
Horse F - equates to 4.76%

The total book percentage is 119.04%.
A bookmaker then knows his "overround" is 19% and that is his theoretical profit margin or edge.

Now the question I have is, if Horse A is a non-runner then by what amount do you reduce the odds of the other runners to get back to a 119% book.

Unfortunately it is not as simple as just reducing each of their odds by 66.67%. That would end up with a book percentage of around 109%.

There must be a basic formula to apply that will reduce each of the remaining runners odds to get back to 119%.

Please can someone help me? Many thanks.

If horse A is a non-runner, then the total percentage before adjustment is: 33.33 + 4.76 + 4.76 + 4.76 + 4.76 = 52.37.

Now adjust all the 5 figures in the following way:

33.33 x 119.04/52.37 = 75.76099
4.76 x 119.04/52.37 = 10.81975
10.81975
10.81975
10.81975

If you now sum up these 5 results, you get 109,04 again.

Repeat the same procedure for any other non-runner or any numbers of non-runners.

Hope this is what you want.

Hi Rassis,
Thank you very much for that answer, which seems to solve our dilemma. I hope our programmer can now use this to create the formula in the code he needs to automate this process. Many thanks for your time. I am very grateful.

what country are you in noel?

In the UK