# 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
The text contains a discussion about the probabilities of race outcomes between Alice, Bob, and Charlie, and asks two specific questions about these probabilities, followed by a generalization to n people. The questions are not answered within the text itself, but a note is provided for context.
multiple_questions
Given a text passage, determine:
    1. If it contains a mathematical question (including word problems or calculations)
    2. If an answer to that mathematical question is provided in the text

First, describe the content structure you found in 1-2 sentences.

Then provide your analysis in the following format:
TYPE: <one of the following values>
- NO_QUESTION
- SINGLE_QUESTION
- SINGLE_QUESTION_WITH_ANSWER
- MULTIPLE_QUESTIONS
- MULTIPLE_QUESTIONS_WITH_ANSWERS
system
# 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

MarkdownAppears as
*italics* or _italics_ italics
**bold** or __bold__ bold
- bulleted- list
• bulleted
• list
1. numbered2. list
1. numbered
2. list
Note: you must add a full line of space before and after lists for them to show up correctly
paragraph 1paragraph 2

paragraph 1

paragraph 2

[example link](https://brilliant.org)example link
> This is a quote
This is a quote
    # I indented these lines
# 4 spaces, and now they show
# up as a code block.

print "hello world"
# I indented these lines
# 4 spaces, and now they show
# up as a code block.

print "hello world"
MathAppears 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} $$\boxed{123}$$

Sort by:

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
user
The text contains a discussion about the probabilities of race outcomes between Alice, Bob, and Charlie, and asks two specific questions about these probabilities, followed by a generalization to n people. The questions are not answered within the text itself, but a note is provided for context.

TYPE: MULTIPLE_QUESTIONS