Math Brainteaser: Probability of Longer Piece Length

[oceanflavored]

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

 

[Dick]

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?

 

[rohanprabhu]

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.

 

[Dick]

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.