Firstly, hello all. I'm normally to be found on StackOverflow but felt that this forum was more appropriate for my question. Count this is a coffee break teaser, rather than a fully challenging maths problem and have fun with it!
Hopefully I can explain this clearly.
The problem
I am trying to establish a weighting for the position of a number within a variable range of elements. The actual element numbers don't matter themeselves, purely their place in the order that they are presented. for example, given the range.
1, 9, 2, 4
the number 9 in simple terms would have a weighting of 0.5 due to my very simple calculation of position/number of items. This of course works great if all my ranges are only comprised 4 items. Things get tricky when for example, I want to derive a weighting for the next range of elements:
1, 46, 2, 9, 23, 4, 5, 56
In this example, the number 9 STILL has a weighting of 0.5 as it conforms to the same equation. What I'd like to achieve is a weighting that signifies the fact that the length of the range is greater and therefore being in the middle (or any other arbitary position) has a radically different index due to the range length etc.
The reason
The rationale behind this is that I'd like to determine an index value for a horse in a race based on its position in a given bookies odds ranking. Therefore, in a field of 20 runners, the horse at position 10 would in theory have the 0.5 index value. However, in a field of 4 runners, the horse ranked #2 (index 0.5) has a far greater chance of winning than the 10th runner of 20. Applying a simple 1 in 20 chance for each runner won't cut it as the ranking is derived from form factors and other shenanigans.
Is there a simple way to represent this requirement? (btw - please feel free to edit the tags, I wasn't 100% certain where my question lay)