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?

#Contradiction #Probability

Easy Math Editor

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Comments

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.

Ivan Koswara - 7 years, 2 months ago

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

Calvin Lin Staff - 7 years, 2 months ago

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....

Pranshu Gaba - 7 years, 2 months ago

Math	Appears as
Remember to wrap math in `\(` ... `\)` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$