Fishing - What's a boy to do?

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What's a boy to do?

"riverman" wrote in news:1162253876.219819.310220
@m7g2000cwm.googlegroups.com:

Remember, we are looking at a conditional probability; dart B has
already landed farther than dart A. So our list of outcomes is limited
to:

ABC
ACB
CAB

Reducing this to a combinatorial problem is incorrect and misleading.
Each dart has a distribution around some point, hopefully the center, with
decreasing probability as you get further from the target. If darts A and
B are three or four standard deviations out, dart C has a very high
probability of being closer to the center. If A and B are a tenth of a
standard deviation out, there is a very low probability of C being closer.
We need to know the 2-D distribution of the dart C, and then we need to
know where the first two darts landed. So "not enough information" is
correct.

--
Scott
Reverse name to reply

What's a boy to do?

Calif Bill wrote:

You could have an almost infinite amount of darts the exact same distance
from the center. The only limiting number is how big the circle is from the
center and how big of diameter is the dart. There are an infinite number of
points equidistant from the center point. And it depends on neither the
precision or accuracy of the measurement. And in your measurement of the
distance it would be more accuracy and not precision. Precision only gives
more numbers after the decimal point.

Bill:
I'm not sure where to start, but there are a lot of little details in
your assertations that are erroneous. There's some truth also, so don't
lose hope :-)

First of all, yes the definition of a circle states there are an
infinite number of points in a plane that are equidistant from a given
point, but the liklihood of getting even two darts to land on that
circle is slim. (Just how slim is discussed in the second paragraph
below.) We don't even have to agree on how slim for now, but the the
more darts you want to have land on that circle, the less likely it is
to happen, and it approaches zero as the number of darts gets larger
and larger. Although the phrase 'almost infinite' is actually
meaningless, I assume you mean we are looking at numbers that are
growing huge beyond comprehension, so the liklihood of it happening is
shrinking tiny beyond comprehension.

Secondly, it IS a matter of precision, not accuracy. We don't care what
the actual distance from the center is, what we do care about is
whether or not two darts have the same measurement from the center,
even if that measurement is wrong. If we use an inaccurate tool, then
we might get a wrong amount (a broken ruler might show each dart to be
10.55 cm from the center, while they are both actually much less that
that). That's 'inaccurate', but if the numbers match, then we can still
assert that they are the same distance. If we use a ruler with really
fat indicator lines, we might get both measuring 10.55 cm, however if
we used a vernier caliper, calibrated or not, we might get one of them
measuring 10.550000000000001 cm and the other measuring
10.550000000000002 cm. Those are measures of high PRECISION, and my
assertation is that, no matter how the darts land, we can always use
more precise measuring devices until we find where the numbers vary.
And they always will, even if we have to go to electron microscope
levels. Just as no two snowflakes are alike, no two darts can land the
same distance from the center.

Now, I appreciate that some people might have an ingrained prejudice
against math because it doesn't always conform to their intuition (and
this might be you, or it might not). But when faced with something that
doesn't seem to 'fit' what we want to believe, there are two choices:
find out the rules of math and learn to analyze things according to
those rules, including learning the constraints and limitations and the
meaning of those, or else continue to assert that what we believe is
right because it 'feels right' to us, and use poorly structured
arguments or misnomers to claim that nothing has any validity, so we
can't possibly be wrong. That way lies madness.

--riverman

What's a boy to do?

Scott Seidman wrote:
...Each dart has a distribution around some point, hopefully the center, with
decreasing probability as you get further from the target.

I detect a troublesome ambiguity here. I see two ways to read this
without the reader doing any violence to good sense:

1. Each dart has a distribution around some point, hopefully the
center, with decreasing probability of being close to the center as the
thrower gets further from the target.

2. Each dart has a distribution around some point, hopefully the
center, with decreasing probability of landing at any point as distance
from the center increases.

I think no one will have much trouble with the first reading. The
second is easier to defend than it may appear at a glance because it
is.....partly.....true. I assume that a truly great dart thower can
consistently place the darts very close to the center of the board
(which, as we all know is not necessarily the object in every game, but
is as good a spot as any other and does, at any rate, appear to be
accepted for the purposes of this discussion) and I know that there are
people whose skills are so abysmal that they rarely hit the board at
all from the standard distance of what I believe to be 8 feet or so.
Probabilities of distribution clearly vary widely (if not to say
wildly) between these two extremes. What makes the second reading
defensible is that for the best dart throwers the probability of
landing at a given point DOES decrease with distance from the
center.....um.....mostly. In fact, it also decreases as the point gets
very very close to a precisely measured center. The same is also true
for ALL dart throwers. The difference among them is that the diameter
of the circle at which increasing or decreasing probabilities converge
or diverge (depending on direction of travel toward or away from the
circle) varies with the skill of the thrower, being very small for the
very good and very large for the very bad.

I haven't looked at this, or the larger discussion, closely enough to
suppose that it will be a crushing blow to anyone's thesis but, on the
other hand, I haven't seen anything yet that I think rules it out
either. It may not even be relevant or interesting given the
assumptions that have been (if only tacitly) agreed on.

However, it does once again open the door to an examination of those
assumptions. Given that no one showed any interest the first time I
brought the matter up though, I guess I won't go into it in any depth
here. I will simply confine myself to making a proposition open to
anyone. Give me three darts and a prediction of where they will land
relative to one another in terms of distance from the center of the
target, and I will prove you wrong EVERY time. :)

Wolfgang

What's a boy to do?

Jonathan Cook wrote:
riverman wrote:

solution. To answer the darts question, merely rely on the definition
of probablilty: the number of ways to achieve your objective, divided
by the number of possible outcomes. List all the possible arrangements
of how the darts could land, and count how many fit our scenario.

I saw your response to my post, but I still disagree. You are making
all sorts of assumptions by reducing it to this simple model, not
the least of which is some sort of uniform probability distribution,
or worse yet, a 50-50 likelihood of landing inside or outside of
dart A. And those assumptions _heavily_ depend on how closely dart
A is to the target, among other things (like if I'm throwing, who
has very little dart-throwing experience). Dart throwing is _nothing_
like dice rolling...

First, list all the ways to throw three darts, A B and C.

ABC
ACB
BAC
BCA
CAB
CBA

Sometime too simple _is_ too simple...

But, Wolfgang's original post has got me thinking, and I'd be interested
to see your response to a post I'm about to make from his original
question (and Wolfgang's and any others' responses, too.)

Jon.

Hi Jon:
Actually, I'm not assuming a uniform distribution, just an unchanging
one. I think the only crucial assumption here (beyond the obvious ones:
gravity is constant, the target is planar, the darts are self-similar,
the wind doesn't change, etc) is that the skill of the thrower does not
improve or deteriorate demonstrably between throws, and that the
thrower is aiming for the same spot with each throw.

Some people are NOT assuming these things, and I agree that it skews
the results, and I'll acquiesce and say that I should have stated those
things and not assumed them. The nonsense about two darts being able to
be equidistant is just that: nonsense. But if we assume the conditions
are identical for all three darts, then each throw is identical without
prejudice, and the six outcomes are equally likely, and the answer is
2/3.

Take this problem across the hall to the Stats professors and see what
they have to say about it.

Your other assertations about probabilities not being meaningful in the
context of singular events sounds bizarre. I'll re-read what you wrote,
mostly because I see you have some education in math and statistics,
but it sounds like you are getting twisted around somewhere. Possibly
within the definition of 'probability'.

--riverman

What's a boy to do?

riverman wrote:
....If we use a ruler with really
fat indicator lines, we might get both measuring 10.55 cm, however if
we used a vernier caliper, calibrated or not, we might get one of them
measuring 10.550000000000001 cm and the other measuring
10.550000000000002 cm. Those are measures of high PRECISION, and my
assertation is that, no matter how the darts land, we can always use
more precise measuring devices until we find where the numbers vary....

If you can REALLY measure things on a scale several orders of magnitude
short of an angstrom, I REALLY wanna come play in your shop! :)

For the benefit of those who don't know what a vernier caliper is (a
friend of mine....who should have known better.....once thought a
micrometer I handed him was some sort of special application c-clamp
and proceeded to see how tight he could crank it.....I nearly smacked
his ****in' hea......well, never mind about that), suffice it to say
that the numbers displayed above were used for effect but Myron's point
is indisputable. Precision measuring instruments these days are such
that finding two darts whose distance from a given point is so close
that the difference can't be measured is vanishingly small. The REAL
difficulty (whether in a pub or a laboratory) would be in getting
agreement on exactly where to measure from.

Wolfgang

What's a boy to do?

"riverman" wrote in message
oups.com...

Calif Bill wrote:

You could have an almost infinite amount of darts the exact same distance
from the center. The only limiting number is how big the circle is from
the
center and how big of diameter is the dart. There are an infinite number
of
points equidistant from the center point. And it depends on neither the
precision or accuracy of the measurement. And in your measurement of the
distance it would be more accuracy and not precision. Precision only
gives
more numbers after the decimal point.

Bill:
I'm not sure where to start, but there are a lot of little details in
your assertations that are erroneous. There's some truth also, so don't
lose hope :-)

First of all, yes the definition of a circle states there are an
infinite number of points in a plane that are equidistant from a given
point, but the liklihood of getting even two darts to land on that
circle is slim. (Just how slim is discussed in the second paragraph
below.) We don't even have to agree on how slim for now, but the the
more darts you want to have land on that circle, the less likely it is
to happen, and it approaches zero as the number of darts gets larger
and larger. Although the phrase 'almost infinite' is actually
meaningless, I assume you mean we are looking at numbers that are
growing huge beyond comprehension, so the liklihood of it happening is
shrinking tiny beyond comprehension.

Secondly, it IS a matter of precision, not accuracy. We don't care what
the actual distance from the center is, what we do care about is
whether or not two darts have the same measurement from the center,
even if that measurement is wrong. If we use an inaccurate tool, then
we might get a wrong amount (a broken ruler might show each dart to be
10.55 cm from the center, while they are both actually much less that
that). That's 'inaccurate', but if the numbers match, then we can still
assert that they are the same distance. If we use a ruler with really
fat indicator lines, we might get both measuring 10.55 cm, however if
we used a vernier caliper, calibrated or not, we might get one of them
measuring 10.550000000000001 cm and the other measuring
10.550000000000002 cm. Those are measures of high PRECISION, and my
assertation is that, no matter how the darts land, we can always use
more precise measuring devices until we find where the numbers vary.
And they always will, even if we have to go to electron microscope
levels. Just as no two snowflakes are alike, no two darts can land the
same distance from the center.

Now, I appreciate that some people might have an ingrained prejudice
against math because it doesn't always conform to their intuition (and
this might be you, or it might not). But when faced with something that
doesn't seem to 'fit' what we want to believe, there are two choices:
find out the rules of math and learn to analyze things according to
those rules, including learning the constraints and limitations and the
meaning of those, or else continue to assert that what we believe is
right because it 'feels right' to us, and use poorly structured
arguments or misnomers to claim that nothing has any validity, so we
can't possibly be wrong. That way lies madness.

--riverman

Your statement was no two darts could land equidistant from the center. As
I state, totally wrong. There is an infinite number of points on the circle
where the first dart landed from the center. You change the statement to
say the more darts that are thrown, the likelihood that two are equidistant
from the center diminishes. The more darts tossed the greater the
likelihood two are equidistant. Not all equidistant, but two can be. Sure,
your vernier calipers can have more precision than a wooden school rule, but
it is still the accuracy of the measurement. Your statement is akin to
saying you have increased precision when you multiply 1.02 times 2.04 and
get 2.0808. You still have only 2 decimal points of precision. As to using
an accurate tool, does not need to be. The actual measurement may be wrong,
but as long as the measurement is the same is all we need to know for the
stated question. As long as the precision and accuracy is great enough in
the measuring instrument to have repeatability, the actual distance matters
not.

Maybe you should go back and review the rules of math. And review your
arguments.

What's a boy to do?

Scott Seidman wrote:
"riverman" wrote in news:1162253876.219819.310220
@m7g2000cwm.googlegroups.com:

Remember, we are looking at a conditional probability; dart B has
already landed farther than dart A. So our list of outcomes is limited
to:

ABC
ACB
CAB

Reducing this to a combinatorial problem is incorrect and misleading.
Each dart has a distribution around some point, hopefully the center, with
decreasing probability as you get further from the target. If darts A and
B are three or four standard deviations out, dart C has a very high
probability of being closer to the center. If A and B are a tenth of a
standard deviation out, there is a very low probability of C being closer.
We need to know the 2-D distribution of the dart C, and then we need to
know where the first two darts landed. So "not enough information" is
correct.

But consider that outcome of throwing each dart is independant. IOW,
there will always be one dart that is closest, one that is farthest,
one that is in between. The combinatoric arrangement merely gives all
the arrangements.

Yes, if Dart A is very very close, then the probability of dart B being
closer gets smaller. But that same distribution gets repeated again
with dart B being closer than A if we throw dart B first. So, unless
the ability of the thrower changes between the two darts, AB is
identical to BA.

--riverman

What's a boy to do?