Place, show and unordered horses forecast probability calculation given the following horses, each with its chance of winning:
Horse 1 -> 0.29
Horse 2 -> 0.34
Horse 3 -> 0.11
Horse 4 -> 0.07
Horse 5 -> 0.14
Horse 6 -> 0.05
Sum -> 1
At the moment, in order to calculate the straight forecast of 1-4, that is the probability that horse 1 wins the race AND horse 2 finishes second, I consider horse 1 as winner and recalculate all the other probabilities excluding horse 1 from recalculation (with proportions). After calculating the probability that horse 4 finishes first on 5 horses total, I multiply the two probabilities. In this way, straight forecast 1-4 and 4-1 are different values. I think this is the correct way to proceed. The same thing I do the for straight tricast, recalculating and multiplying for two horses instead of one.
But I don't know how to calculate:


*

*the probability that horse 1 finish first OR second (place bet)

*the probability that horse 1 finish first OR second OR third (show bet)

*the probability that result is 1-4 OR 4-1 (reverse forecast)

*the probability that result is 1, 4 and 3 finish in any order (combination tricast)


I don not simply need a formula, but a full mathematical explanation in order to fully understand. Academic papers on Internet regarding this argument are well accepted.
 A: This is fine if you are dealing with dice or a specific cards out of a deck. Unfortunately, when handicapping horses,  The odds of  winning do not necessarily reflect the odds of finishing second or third. I love calculating odds, but there are reasons this will not work with horses.   
Imagine a 10 horse race where the odds of winning on numbers 1 and 2 are both 5-2. Horses 3 and 4 are both 7 to 1. 
This would make one think that the odds of the exact of being 13, 14, 23, and 24 would be equal. 
However, this is horse racing. What if the numbers 1 and 3 are speed horses, and numbers 2 and 4 are very late running closers. 
1 and 3 want a slow pace. 2 and 4 need a fast pace. 
13 has a better chance than 14, and 24 better than 23. 
With horses, the pace can make things great for a horse or knock him out of contention, and often, if you tell me a certain horse is going to win, I can  say with greater accuracy which horse is going to be second. If you told me that the 2 horse had won this race, and I knew nothing else aside from the 1&3 horses being speed horses and the 2&4  being closers,  odds are the 4 had a better chance for a second then the 3. 
This is where handicapping comes into play as opposed to using pure mathematics in casino games.
