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The way I see it, is that anything less than 1 cubed equals less than 1, anything over 1 cubed equals greater than 1. There are equal numbers between 0 and 1 to 1 and 2 so the percentage of numbers cubed below 1 (including 1) which will equal 1 or less is 50% and the percentage of numbers between 1 and 2 which will equal a cubic volume between 1 and 2 is also 50%.

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My views do not represent Safespeed but those of a driver who has driven for 39 yrs, in all conditions, at all times of the day & night on every type of road and covered well over a million miles, so knows a bit about what makes for safety on the road,what is really dangerous and needs to be observed when driving and quite frankly, the speedo is way down on my list of things to observe to negotiate Britain's roads safely, but I don't expect some fool who sits behind a desk all day to appreciate that.

8 cubic centimeters is a cube with 2 cm per side.

A cube of 1 cubic cm is a cube with 1 cm per side.

The answer is 50%

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The views expressed in this post are personal opinions and do not necessarily represent the views of Safe Speed.
You will be branded a threat to society by going over a speed limit where it is safe to do so, and suffer the consequences of your actions in a way criminals do not, more so than someone who is a real threat to our society.

But the number of numbers between 0 and 0.1 is equal to the number of numbers between 1 and 2 and is also equal to the number of numbers between 1 and five thousand million: its is \aleph_1 - the cardinality of the set of all countable ordinal numbers. As an infinity is not a number doing arithmetic with it, as you do, can lead to all kinds of contradictions, such as the famous proof that 1 = 0.

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When I see an adult on a bicycle, I do not despair for the future of the human race. H.G. Wells
When I see a youth in a motor car I do d.c.brown

8 cubic centimeters is a cube with face area 4 cm^2.
A cube of 1 cubic cm is a cube with face area 1 cm^2
The answer is 25%

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When I see an adult on a bicycle, I do not despair for the future of the human race. H.G. Wells
When I see a youth in a motor car I do d.c.brown

Read the question again. What is the PROBABILITY of its VOLUME being 1cc or less?

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My views do not represent Safespeed but those of a driver who has driven for 39 yrs, in all conditions, at all times of the day & night on every type of road and covered well over a million miles, so knows a bit about what makes for safety on the road,what is really dangerous and needs to be observed when driving and quite frankly, the speedo is way down on my list of things to observe to negotiate Britain's roads safely, but I don't expect some fool who sits behind a desk all day to appreciate that.

As long as dcb doesn't get it now, I'm happy

_________________
The views expressed in this post are personal opinions and do not necessarily represent the views of Safe Speed.
You will be branded a threat to society by going over a speed limit where it is safe to do so, and suffer the consequences of your actions in a way criminals do not, more so than someone who is a real threat to our society.

I thought Pete already said the answer is 50%. Grabby got it right. We're looking for a paradox now aren't we?

_________________
The views expressed in this post are personal opinions and do not necessarily represent the views of Safe Speed.
You will be branded a threat to society by going over a speed limit where it is safe to do so, and suffer the consequences of your actions in a way criminals do not, more so than someone who is a real threat to our society.

I don't think that Pete has given an answer yet. And there are no mathematical paradoxes: just bad mathematicians

The flaw with your reasoning is that the problem is stated in terms of volume not edge length. And there are seven times as many possible volumes for cubes with edges between one and two than for cubes with edges less than one.

I have just done a simulation on this problem and, with a sample size of 10^7, get answers between 12.48% and 12.51%

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When I see an adult on a bicycle, I do not despair for the future of the human race. H.G. Wells
When I see a youth in a motor car I do d.c.brown

Keep up dcb

_________________
The views expressed in this post are personal opinions and do not necessarily represent the views of Safe Speed.
You will be branded a threat to society by going over a speed limit where it is safe to do so, and suffer the consequences of your actions in a way criminals do not, more so than someone who is a real threat to our society.

Of course the units matter. Volume is length cubed. Area is length squared.

If the question is about edge length the answer is 1/2 50%
If the question is about face area the answer is 1/2^2 25%
If the question is about volume the answer is 1/2^3 12.5%.

What was the question again?

If you generate 1000 random numbers in the range 0 to 8, representing a 1000 random cubes you wil find ~125 of them are less than one.

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When I see an adult on a bicycle, I do not despair for the future of the human race. H.G. Wells
When I see a youth in a motor car I do d.c.brown

Ok, it seems that I came close to setting off WWIII here so here goes:

If the sides of the cubes are in integer units of 1cm, the only two possible cubes are 1 and 8 cu cm. You cannot have, for example, a 5 cu cm cube,
If the sides are in mm, you can have 20 possible cubes, with 20 possible volumes.
The same holds true for whatever units you choose, right down to units of one atom, or molecule, or whatever the cubes are made of. You cannot have a cube consisting of 4 atoms, for example. (you could go to even smaller units theoretically, but not physically)
So, the number of possible cubes of less than or equal to 1 cu cm is equal to the number of possible cubes equal to or less than 8 cu cm but greater than 1 cu cm.

I know I'm not very good at explaining things, but I hope this is clear enough

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Only when ideology, prejudice and dogma are set aside does the truth emerge - Kepler

However, the probability side of the problem is lacking. You seem to be assuming that all distributions are uniform.

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http://kalvis.com/

When I said random I assumed a uniform distribution.
The thing is, you can't have a uniform distribution of volumes. If you have a cube with sides of a trillion trillion light years, and you remove just one atom from it, you no longer have a cube.

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Only when ideology, prejudice and dogma are set aside does the truth emerge - Kepler

Random does not mean uniform, so add the words "uniform distribution" next time. And you can have a uniform distribution of volumes.

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http://kalvis.com/

Here is a new question, and here I am assuming a uniform distribution.

Take a finite line element (or thin rod, say), and cut it in two random places, assuming independence, and find the probability that you can make a triangle out of the pieces.

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http://kalvis.com/

Finite elements are a subset of continuous variables. I assumed a continuous variable (the length) as nothing was specified, simply because it is the way of the world (until you reach the level of quantum mechanics). So I would say my conclusion was correct and for the right reason - but I would say that

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Views expressed are personal opinions and are not necessarily shared by the Safe Speed campaign

Can this be reduced to "what is the probability that the sum of two numbers is equal or greater than a single number (all from a uniform distribution)"?

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Views expressed are personal opinions and are not necessarily shared by the Safe Speed campaign

I will tell you, when you provide an answer! Remember the two cuts are in the same line element!

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http://kalvis.com/

That's one way of looking at it.
Another way is that no matter how many times you subdivide integral units, you always end up with yet smaller integral units.
Continuous variables imply infinity - which only exists in the abstract.

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Only when ideology, prejudice and dogma are set aside does the truth emerge - Kepler

Given that any side of the triangle cannot be greater than the sum of the other two sides, I'd say that the probability is 0.5

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Only when ideology, prejudice and dogma are set aside does the truth emerge - Kepler