# Probabilistic Race Results

Consider a race that is run by several people, including Alice, Bob and Charlie.

1) Is it possible that the probability that Alice finishes before Bob is strictly greater than $$\frac{1}{2}$$ AND the probability that Bob finishes before Alice is strictly greater than $$\frac{1}{2}$$?

2) Is it possible that the probability that Alice finishes before Bob is strictly greater than $$\frac{1}{2}$$ AND the probability that Bob finishes before Charlie is strictly greater than $$\frac{1}{2}$$ AND the probability that Charlie finishes before Alice is strictly greater that $$\frac{1}{2}$$?

Can we generalize this to having $$n$$ people?

Note by Calvin Lin
4 years, 2 months ago

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If Alice finishes before Charlie, and Charlie finishes before Bob, i will be denoting it as $$\textrm{ACB}$$. I will be assuming that no two people can finish together.

1) Let's assume both $$P(\text{AB}), P(\text{BA})$$, are both strictly greater than $$\dfrac{1}{2}$$, then this means $$P(\text{AB}) + P(\text{BA}) > 1$$.

However, we know that $$\text{AB }$$and $$\text{BA}$$ are mutually exclusive events, and collectively exhaustive, therefore $$P(\text{AB}) + P(\text{BA}) =1$$. This is a contradiction, therefore our initial assumption was wrong, which means, both $$P(\text{AB})$$ and $$P(\text{BA})$$ cannot be strictly greater than $$\dfrac{1}{2}$$

2) I am working on this right now....

- 4 years, 2 months ago

I suppose this problem comes from nontransitive dice, although by making it a scenario of runners my thoughts of probability are suddenly shattered. Still figuring out how to formalize the probabilities as runners.

- 4 years, 2 months ago

That is a good interpretation, and gives you a possible construction almost immediately.

Staff - 4 years, 2 months ago