A long straw is tossed at random onto a circle; given that it falls so that it intersects the circle, what...

In article <38mpvq$1r5@agate.berkeley.edu>, isaackuo@OCF.Berkeley.EDU
(Isaac Ji Kuo) writes:
> [I wrote]
> >Why is it important to answer puzzles as they are posted?
>
> Because anything not stated is not stated. Give us one reason
> to be restricted by rules clearly not stated in the problem.
>
> You mention a solution in another posting claiming to remove the
> ambiguity by refering to "The Well Posed Problem."  Well, it's too bad
> that that solution is a solution to a _well_ posed problem.  So what if
> that other problem is well posed, and not ambiguous?  It in no way
> makes the problem you posted here in this thread less ambiguous.

The well posed problem says:

'A long straw is tossed at random onto a circle; given that it falls so
that it intersects the circle, what is the probability that the chord
thus defined is longer than the side of an equilateral triangle?'

In what way is this less ambiguous than the version I posted?

You seem to have missed the point in the answer - the problem said
nothing about the size of the circle therefore the size of the circle
does not matter.  It said nothing about the position of the circle
therefore the position of the circle does not matter.

Given these two implicit pieces of information, it is only possible to
get ONE random distribution that gives the same answer for all circles,
therefore this MUST be the answer.

-- Paul McCombes <paul@mccombes.demon.co.uk>

[ Editor: the collected posting does not state the resulting
probability, and the earlier postings it refers to were not
collected. ]

[ Editor: further material on this puzzle, edited separately from other collected postings (batch 24); it may repeat some of the above. ]

[ Editor: problem as stated there: ]
What are the different ways of defining a "random chord" of a circle?

[ Editor: problem statement reconstructed from the posting, which
answers "Anyone have other methods of defining a random chord?" in a
1997 thread titled "Crossed Cords", whose original question is not in
the collected material. ]


> I assume that the answer you are looking for is 1/3.
>
> I believe the odds are different for each "random"chord method
> chosen.  Anyone have other methods of defining a random chord?

-- Boston3455 <boston3455@aol.com>

From an old post of mine...

As pretty much everyone here now knows, this business of "random
chords" in the circle merely depends on the precise definition of
"random" in this context, which is not quite as canonical and automatic
as "random" choices in some other contexts.

Taking (as usual) a "random choice" of a number from a finite
interval/area to mean uniform, and more than one such "random" choice
to mean independent, the "big three" definitions usually given are the
first three given in the list below; but as you can see these are not
the only candidates that could be adopted!  The full set of
possibilities I could think of are...

1. Fix a diameter; choose a random point along it.
   Draw the perpendicular chord.

1A. Ditto with a random diameter; angle chosen in (0,2pi).

2. Choose a random point in the interior of the circle.
   Draw the perpendicular bisector to the radius so defined.

3. Fix a circumference point.  Choose a random direction from it
   (0,2pi).  Draw the chord so defined.

3A. Ditto with a random point on the circumference.

4. Choose a random point in the interior of the circle.
   Choose a random direction from it (0,2pi).  Draw the chord so
   defined.

5. Choose two random points in the interior of the circle.  Join them.

6. Choose two random points on the circumference of the circle.  Join
   them.

6A. Ditto with one of the points fixed in advance.

7. Choose a random point in the interior of the circle.
   Choose a random point on the circumference of the circle.  Join
   them.

7A. Ditto with a fixed circumference point.

8. Fix a direction, (e.g. NS).  Choose an interior point at random.
   Draw the chord in the direction, through the point.

9. Fix a direction, (e.g. NS).  Choose a circumference point at random.
   Draw the chord in the direction, through the point.

Well that's unlucky 13 different possibilities!  Though 1A 3A 6A & 7A
are pretty obviously the same as their plain numbers.  And a moment's
further thought shows that similarly, 8 is really 4A, and 9 is really
3A-A (!)  Still that leaves seven possiblities; perhaps six if you want
to drop the slightly mixed and thus fishy-looking #7.

More interestingly, #6(A) is actually the same as #3(A); though the
proof needs a trite Euclidean geometry observation.  (There may well be
other equivalent pairs in there - haven't checked fully.)  But that
*still* leaves five or six random-chord definitions, all with
(presumably) different probability distributions, medians, means and
other features.

And finally, here's a rather vaguer method:-

10. Take a huge circle with a fixed circle in the middle.  Choose a
    random chord of this huge circle, (it shouldn't matter which way!),
    repeating the choice until the chosen chord intersects the inner
    circle.  Now let huge --> infinity.

So which is the "right" one?

-- Bill Taylor <wft@math.canterbury.ac.nz>