Statistical Relationship between Win and Place odds

[Arbu]

I'd like to develop an understanding of how win and place odds are related mathematically. Essentially, from the win odds on betfair can you calculate the theoretical place odds? I'd see it working something like as follows. Suppose two horses are to place. If you look at one horse, you know its odds of winning from the win odds. So its chances of not winning are 1 - this. You could then look at the chances of it being beaten by each of the other horses in the race and then see the odds of it coming second in a set of races constituted of the remaining horses. Then add up all the odds of it winning those reconstituted races with the odds of it winning to get the place odds. Anyone have any input on this?