A rod (length = 1 unit) is broken into two pieces with the breakage point
being a uniform distribution (i.e. each point being equally likely).

What is the probability that the lengths of the two pieces have a rational
ratio?

My answer: 0.

The ratio must always be irrational.

--
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On 25/12/2021 00:08, Diego Garcia wrote:
> A rod (length = 1 unit) is broken into two pieces with the breakage point
> being a uniform distribution (i.e. each point being equally likely).
>
> What is the probability that the lengths of the two pieces have a rational
> ratio?
>
> My answer: 0.
>
> The ratio must always be irrational.
>

Assuming you mean a mathematical rod, representing a real number interval of length 1, so this is a
mathematical probability question, then I agree. The set of rational numbers has measure zero as it
is countable, so P(rational breakage point) = 0.

Put differently, if q0, q1, q2, ... is an enumeration of all the rational breakage points, then

P(rational breakage point) = P( {q0, q1, q2, ...} )
= P( {q0} ) + P( {q1} ) + P( {q2} ) + ...
= 0 + 0 + 0 + ...
= 0

For the second equality - this holds because there are /countably/ many disjoint events {q0}, {q1},
{q2} ..., so the probability of their union is the sum of their individual probabilities.

To say "the interval must always be irrational" might confuse people. After all, this isn't some
actual physical experiment that you're going to perform! Rational breakage points are VALID
breakage points, but it's just that the probability of all such breakage points is zero. The phrase
"the breakage point is /almost certainly/ irrational" is the usual phrase for this.

https://en.wikipedia.org/wiki/Almost_surely

Mike.

On 12/24/2021 6:08 PM, Diego Garcia wrote:
> A rod (length = 1 unit) is broken into two pieces with the breakage point
> being a uniform distribution (i.e. each point being equally likely).
>
> What is the probability that the lengths of the two pieces have a rational
> ratio?
>
> My answer: 0.
>
> The ratio must always be irrational.
>
>

when one measures a length, there is a rational number you can construct that can be so close to it, there is near 0 difference.

so the length is rational, so then the ratio is rational.

Serg io wrote:

> when one measures a length, there is a rational number you can construct
> that can be so close to it, there is near 0 difference.
> so the length is rational, so then the ratio is rational.

American indian chiefs selling manhattan to the white newcomers for beads
worth $24. Gorby was the indian chief in his relations with the west.
Only more stupid, He did not even get a $20 of beads in return. Worse.
Opened the doors to Russia for a fake friendly pat on the shoulder.

Gorbachev should have faced trail for for treachery and crimes against
the Soviet state. Policies which has now lead to Russia being encircled
by NATO after believing the lying hyenas. So many people got poor and
hungry, or died in wars in countries which was the result of this
betrayal of the Motherland.

On Sat, 25 Dec 2021 00:53:07 +0000, Mike Terry wrote:

>
> Assuming you mean a mathematical rod, representing a real number interval of length 1,
>

Of course.

In order to apply mathematics to the physical world the continuity assumption
must hold.

With SI units (length in meters), the continuity assumption regarding matter begins
to break down at 9-10 decimal places.

But, AFAIK, we cannot even measure lengths with an accuracy that exceeds 4-5
decimal places and so for ordinary purposes continuity does hold.

>
> Rational breakage points are VALID breakage points
>

We can perhaps consider a perfect monatomic crystal. Such a substance
can only have rational lengths that are multiples of the inter-atomic distance.
But perfect crystals are rare and most matter is highly irregular and amorphous
and better permits the continuity assumption.