Probability of one horse finishing ahead of another In a 5 horse race.  Lets say the probability of a horse to win the race are as follows:
Horse A : 25%
Horse B : 4%
Horse C : 35%
Horse D : 10%
Horse E : 26%
How would you determine the probability of Horse E finishing the race in front of Horse C ? i.e. (Horse E could come 4th and Horse C 5th).
Is there a method for determining the probability given the information available ?

This is similar to Christian Hennig's answer.
If you make the strong assumption that the probabilities are in effect weights, and the first horse is sampled with probabilities proportional to the weights of all the horses, the second horse sampled with probabilities proportional to the weights of all the horses except that already selected as the first horse, and so on, then the answer to your question is simple, namely $$\frac{P(E)}{P(E)+P(C)}= \frac{26\%}{26\%+35\%}=\frac{26}{61}\approx 0.426$$ since, if at any stage neither E nor C have been sampled yet and one of them is sampled at that stage, the probability that it is E is $\frac{26}{61}$ and that it is C is $\frac{35}{61}$.   Other assumptions about horses which do not come first finish would produce different results.
This is how R's sample() function does weighted samples without replacement, so it is easy to simulate, for example with
positions <- function(probs){  
  h <- names(probs)
  result <- sample(h, prob=probs)
  c(which(result == h[1]), which(result == h[2]), which(result == h[3]), 
    which(result == h[4]), which(result == h[5]))  #  positions in simulation
  }

set.seed(2023)
probsABCDE <- c("A"=0.25, "B"=0.04, "C"=0.35, "D"=0.10, "E"=0.26)
sims <- replicate(10^5, positions(probsABCDE))
rownames(sims) <- names(probsABCDE)
rowMeans(sims == 1) # who comes first
#       A       B       C       D       E 
# 0.24825 0.04160 0.34847 0.09984 0.26184 

which is close to the original probabilities allowing for simulation noise.
Actually addressing the question of the probability of Horse E finishing the race in front of Horse C, the simulated probability is close to the theoretical probability allowing for simulation noise:
mean(sims["E",] < sims["C",])
# 0.42862
probsABCDE["E"] / (probsABCDE["E"] + probsABCDE["C"]) 
# 0.4262295

