Math Brainteaser: Probability of Longer Piece Length

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Homework Help Overview

The problem involves determining the probability that one piece of a string, cut at a random point, is at least x times as long as the other piece. It falls within the subject area of probability theory, specifically focusing on continuous probability distributions.

Discussion Character

  • Exploratory, Conceptual clarification, Mathematical reasoning

Approaches and Questions Raised

  • The original poster expresses confusion regarding the nature of the problem, noting that traditional probability often involves discrete choices. Participants suggest starting with a simpler case to understand the concept better and question how to measure the favorable outcomes against the total possible outcomes.

Discussion Status

Participants are engaging in a productive discussion, with some offering guidance on how to approach the problem by considering the ratio of lengths in a continuous context. There is an acknowledgment of the need for the original poster to share their attempts to facilitate further assistance.

Contextual Notes

There is a mention of the distinction between discrete and continuous probability distributions, highlighting the complexity of the problem due to the infinite points available for cutting the string.

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Homework Statement


A piece of string is cut into two pieces at a randomly selected point. What is the probability that the longer piece is at least x times as long as the shorter piece?

Homework Equations


none

The Attempt at a Solution


everyone in my family tried, but we couldn't figure it out.
this problem is strange, because usually probability deals with picking something out of a sum total.

thanks for any help :biggrin:
 
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Your family trying doesn't count so much as you trying. What did you try? Try a simple case first. What's the probability you cut the string in such a way that one piece is at least twice the length of the other? What's the measure of places where you can cut the string so it works versus the total measure of places on the string?
 
Last edited:
oceanflavored said:
usually probability deals with picking something out of a sum total.

That is the case when we deal with Discrete Probability distribution. In this case, as you can see, there are infinite points to chose from, there isn't a set of finite points that are favorable. Rather, there is a 'range' of lengths that is favorable. As such, you are dealing with a Continuous Probability distribution.

and as Dick said already, we can't help you further unless you show us something.
 
Continuous versus discrete really aren't that different. And rohanprabhu is absolutely correct. The probability is the ratio of the length of the segments of points that work over the length of the whole segment. I sort of regret using the word 'measure' instead of 'length'. But it's really the same thing as taking the ratio of counts.
 

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