Predicting Race Results knowing winning probability

Question

I would like to model a scenario where I have 4 runners [A, B, C, D] and there chances of winning a race are deemed to be [0.5, 0.3, 0.15, 0.05].

I would like to simulate this race 100 times.

Is it correct that I use Poisson Distribution to calculate the probability of each runner finishing in each of the 4 positions? But then how do I combine these to produce a simulated Race Result?

I would like to achieve an output in the format of:

        [1st, 2nd, 3rd, 4th]
Race 1: [ A,   C,   B,   D ]
Race 2: [ B,   A,   C,   D ]
...