U-boats and Pop-Sci ??

Discussion in 'Health and medical' started by knowknot, Dec 30, 2003.

  1. knowknot

    knowknot Guest

    Would some statistically-knowledgeable folk be good enough to explain the mathematical/statistical
    reasoning (or, if applicable, the principles which contrave) the following statement by Nobel
    Physicist Steven Weinberg (writing about books on war, not about physics) in the most recent (Nov.
    6, 2003) issue of the "New York Review of Books"?

    Prof. Weinberg writes:

    "It should have been obvious that the solution to the U-boat threat was to require merchant ships
    to sail in convoy. As Churchill later explained in The World Crisis, The size of the sea is so
    vast that the difference between the size of a convoy and the size of a single ship shrinks in
    comparison almost to in- significance. There was in fact nearly as good a chance of a convoy of
    forty ships in close order slipping unperceived between the patrolling U-boats as there was for a
    single ship; and each time this happened, forty ships escaped instead of one. (This is also the
    reason that fish of many species swim in schools.)"

    Putting aside the at best highly questionable (and "scientific"?) parenthetical throw-away remark
    about "the reason" ascribed to what "many" fish do, and also disregarding for the moment the
    variable of the role of spying/intelligence, does the Churchill quotation really (accurately)
    "explain" what Weinberg characterizes as "obvious"?

    Does Churchill's (and, implicitly, Weinberg's) use of "vast" conflate that word with "infinite" and,
    conversely, is "vast" itself helpful bearing in mind that, even if a particular shipping route with
    respect to one specific ship (or one convoy) at one time is not known in advance, the geographical
    parameters of British (or of U.S. or other "allied") shipping routes was (more or less) reasonably
    ("probably"?) predictable?

    (And, BTW, what might Weinberg have been referring to by his parenthetical reference to
    "many" fish?)

    Thanks.
     
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  2. Anon.

    Anon. Guest

    knowknot@mindspring.com wrote:
    > Would some statistically-knowledgeable folk be good enough to explain the mathematical/statistical
    > reasoning (or, if applicable, the principles which contrave) the following statement by Nobel
    > Physicist Steven Weinberg (writing about books on war, not about physics) in the most recent (Nov.
    > 6, 2003) issue of the "New York Review of Books"?
    >
    > Prof. Weinberg writes:
    >
    > "It should have been obvious that the solution to the U-boat threat was to require merchant
    > ships to sail in convoy. As Churchill later explained in The World Crisis, The size of the sea
    > is so vast that the difference between the size of a convoy and the size of a single ship
    > shrinks in comparison almost to in- significance. There was in fact nearly as good a chance of a
    > convoy of forty ships in close order slipping unperceived between the patrolling U-boats as
    > there was for a single ship; and each time this happened, forty ships escaped instead of one.
    > (This is also the reason that fish of many species swim in schools.)"
    >
    > Putting aside the at best highly questionable (and "scientific"?) parenthetical throw-away remark
    > about "the reason" ascribed to what "many" fish do, and also disregarding for the moment the
    > variable of the role of spying/intelligence, does the Churchill quotation really (accurately)
    > "explain" what Weinberg characterizes as "obvious"?
    >
    It all depends. If a U-boat can only sink one ship per contact, then this makes sense. If it can
    sink several, then it may or may not make sense, depending on the distribution of the numbers of
    ships it can sink.

    > Does Churchill's (and, implicitly, Weinberg's) use of "vast" conflate that word with "infinite"
    > and, conversely, is "vast" itself helpful bearing in mind that, even if a particular shipping
    > route with respect to one specific ship (or one convoy) at one time is not known in advance, the
    > geographical parameters of British (or of U.S. or other "allied") shipping routes was (more or
    > less) reasonably ("probably"?) predictable?
    >
    Now you're bringing reality into the picture. :)

    This is something that needs to be examined carefully before coming to some conclusion. Actually,
    someone probably has done this, in the context of foraging.

    Bob

    --
    Bob O'Hara

    Rolf Nevanlinna Institute
    P.O. Box 4 (Yliopistonkatu 5) FIN-00014 University of Helsinki Finland Telephone: +358-9-191 23743
    Mobile: +358 50 599 0540 Fax: +358-9-191 22 779 WWW: http://www.RNI.Helsinki.FI/~boh/
     
  3. Brett Aubrey

    Brett Aubrey Guest

    <knowknot@mindspring.com> wrote in message
    news:bnn5fs$ekl$1@darwin.ediacara.org...
    > Would some statistically-knowledgeable folk be good enough to explain the mathematical/statistical
    > reasoning (or, if applicable, the principles which contrave) the following statement by Nobel
    > Physicist Steven Weinberg (writing about books on war, not about physics) in the most recent (Nov.
    > 6, 2003) issue of the "New York Review of Books"? Prof. Weinberg writes: "It should have been
    > obvious that the solution to the U-boat threat was to require merchant ships to sail in convoy. As
    > Churchill later explained in The World Crisis, The size of the sea is so vast that the difference
    > between the size of a convoy and the size of a single ship shinks in comparison almost to
    > insignificance. There was in fact nearly as good a chance of a convoy of forty ships in close
    > order slipping unperceived between the patrolling U-boats as there was for a single ship; and each
    > time this happened, forty ships escaped instead of one. (This is also the reason that fish of many
    > species swim in schools.)"
    >
    > Putting aside the at best highly questionable (and "scientific"?) parenthetical throw-away remark
    > about "the reason" ascribed to what "many" fish do, and also disregarding for the moment the
    > variable of the role of spying/intelligence, does the Churchill quotation really (accurately)
    > "explain" what Weinberg characterizes as "obvious"?

    This won't be the statistical or scientific answer you wanted, but yes is my SWAG-ish answer, and
    the important Churchill quote is: "the difference between the size of a convoy and the size of a
    single ship shrinks in comparison almost to insignificance."

    > Does Churchill's (and, implicitly, Weinberg's) use of "vast" conflate that word with "infinite"
    > and, conversely, is "vast" itself helpful bearing in mind that, even if a particular ship- ping
    > route with respect to one specific ship (or one convoy) at one time is not known in advance, the
    > geographical parameters of British (or of U.S. or other "allied") shipping routes was (more or
    > less) reasonably ("probably"?) predictable?

    No, no conflation. "Vast" is somewhat superfluous, and may therefore only be of help to the
    individual reader. The previous responses are true simply because we know the far-less-than-infinite
    parameters of the North Atlantic. Ironically, the latter half of your comment, above, makes me think
    that you may not appreciate how "vast" is the North Atlantic when compared to one specific ship (or
    one convoy), the permutations for crossing it, or the difficulties with visual acquisition. Remember
    that, especially in the case of World War II U-boats, the conning tower (or fin, depending on from
    whence you come), was anything but lofty - the horizon was not *that* far away - by my rough
    calculations, a submarine on the surface could visually miss a convey a mere ~20 kms (~13 miles)
    distant in fair conditions. This on an ocean with routes spanning a couple of thousand miles. Also
    important is timing, as the pair (boat/convoy, or boat/ship) generally needed to be in proximity
    during daylight hours and in reasonable visibility. And of course the bottom (or retrospective) line
    is that, it worked.

    > (And, BTW, what might Weinberg have been referring to by his parenthetical reference to "many"
    > fish?) Thanks.

    Unclear on your question... "many species" were implied, not many fish. Are you asking which species
    school to minimize predation? There are many, of course, though this can be couterproductive as some
    predators such as certain shark species, marlin, etc., have "learned" (instinct?) well to hover
    above, below and around some schooling fish and periodically and repeatedly race through the school
    to eat their fill - sometimes several predators will quite literally decimate a school together.
    Best regards, Brett.
     
  4. <knowknot@mindspring.com> wrote in message
    news:bnn5fs$ekl$1@darwin.ediacara.org...
    > Would some statistically-knowledgeable folk be good enough to explain the mathematical/statistical
    > reasoning (or, if applicable, the principles which contrave) the following statement by Nobel
    > Physicist Steven Weinberg (writing about books on war, not about physics) in the most recent (Nov.
    > 6, 2003) issue of the "New York Review of Books"?
    >
    > Prof. Weinberg writes:
    >
    > "It should have been obvious that the solution to the U-boat threat was to require merchant
    > ships to sail in convoy. As Churchill later explained in The World Crisis, The size of the sea
    > is so vast that the difference between the size of a convoy and the size of a single ship
    > shrinks in comparison almost to in- significance. There was in fact nearly as good a chance of a
    > convoy of forty ships in close order slipping unperceived between the patrolling U-boats as
    > there was for a single ship; and each time this happened, forty ships escaped instead of one.
    > (This is also the reason that fish of many species swim in schools.)"
    >
    > Putting aside the at best highly questionable (and "scientific"?) parenthetical throw-away remark
    > about "the reason" ascribed to what "many" fish do, and also disregarding for the moment the
    > variable of the role of spying/intelligence, does the Churchill quotation really (accurately)
    > "explain" what Weinberg characterizes as "obvious"?
    >
    > Does Churchill's (and, implicitly, Weinberg's) use of "vast" conflate that word with "infinite"
    > and, conversely, is "vast" itself helpful bearing in mind that, even if a particular shipping
    > route with respect to one specific ship (or one convoy) at one time is not known in advance, the
    > geographical parameters of British (or of U.S. or other "allied") shipping routes was (more or
    > less) reasonably ("probably"?) predictable?
    >
    > (And, BTW, what might Weinberg have been referring to by his parenthetical reference to
    > "many" fish?)

    Churchill's characterization is correct. You can test it yourself if you have a dart board handy.

    Stick a 3-inch flathead nail on the dart board. Now try to hit it with a dart from the regulation
    darting distance (4 meters?). Then replace the nail with a pin and try to hit it with a dart. The
    difference in the frequency of hits (probably near zero in both cases) is insignificant.

    This same phenomanon is what drives fish to form schools. An individual has less chance of being
    eaten by one of its larger, but much less numerous predators when swimming in a school than by
    itself dispersed in the ocean.

    Frank

    >
    > Thanks.
     
  5. joe

    joe Guest

    In article <bnn5fs$ekl$1@darwin.ediacara.org>,
    <knowknot@mindspring.com> wrote:
    >Would some statistically-knowledgeable folk be good enough to explain the mathematical/statistical
    >reasoning (or, if applicable, the principles which contrave) the following statement by Nobel
    >Physicist Steven Weinberg (writing about books on war, not about physics) in the most recent (Nov.
    >6, 2003) issue of the "New York Review of Books"?
    >
    >Prof. Weinberg writes:
    >
    > "It should have been obvious that the solution to the U-boat threat was to require merchant ships
    > to sail in convoy. As Churchill later explained in The World Crisis, The size of the sea is so
    > vast that the difference between the size of a convoy and the size of a single ship shrinks in
    > comparison almost to in- significance. There was in fact nearly as good a chance of a convoy of
    > forty ships in close order slipping unperceived between the patrolling U-boats as there was for a
    > single ship; and each time this happened, forty ships escaped instead of one. (This is also the
    > reason that fish of many species swim in schools.)"
    >
    >Putting aside the at best highly questionable (and "scientific"?) parenthetical throw-away remark
    >about "the reason" ascribed to what "many" fish do, and also disregarding for the moment the
    >variable of the role of spying/intelligence, does the Churchill quotation really (accurately)
    >"explain" what Weinberg characterizes as "obvious"?

    I'm not surprised at Churchill's misunderstanding (or misrecollection) of the convoy issue, but I do
    hope that the Weinberg citation was aimed at answering a different question.

    While yes, a (very) close-order convoy may have nearly the same probability of detection as a single
    ship, if the U-boats then sank all of them once a convoy was detected, the average fraction of all
    ships sunk would be the same either way. If the U-boat was alone and only had time to sink one or
    two boats of the convoy, there would be some protective effect. In reality multiple U-boats were
    steered to convoys once those were detected, and wolf-packs formed (and their codes were broken, and
    the story gets more and more complicated).

    A major reason for convoy sailing was not probability of detection but the defense of the convoy, as
    one escort vessel could protect multiple ships, and it would be impossible to escort all ships if
    they were sailing individually. I don't think this the issue with fish.

    Incidentally, there is an evolutionary connection with convoy issues (aside from the school-of-fish
    connection). C. H. Waddington, a noted geneticist, developmental biologist and evolutionist, headed
    an operations research group in World War II as part of RAF Coastal Command. He wrote a book about
    these issues:

    Waddington, C. H. 1973. OR in World War 2: Operational Research against the U Boat, P. Elek
    (Scientific Books) Ltd.

    My postdoctoral advisor, the late Alan Robertson, was a physical chemistry graduate student at
    Cambridge University who joined Waddington's RAF group. At the end of the war Waddington recruited
    Robertson to join him as a member of his Institute of Animal Genetics at the University of
    Edinburgh. That was good judgement on the part of "Wad", and the start of Alan's brilliant work on
    the theory of quantitative genetics.

    --
    Joe Felsenstein joe@removethispart.gs.washington.edu Department of Genome Sciences, University of
    Washington, Box 357730, Seattle, WA 98195-7730 USA
     
  6. Brett Aubrey

    Brett Aubrey Guest

    <joe@removethispart.gs.washington.edu> wrote in message
    news:bnpnuo$15hh$1@darwin.ediacara.org...
    > In article <bnn5fs$ekl$1@darwin.ediacara.org>, <knowknot@mindspring.com> wrote:
    > >Would some statistically-knowledgeable folk be <snip>...
    > I'm not surprised at Churchill's misunderstanding (or misrecollection) of the convoy issue, but I
    > do hope that the Weinberg citation was aimed at answering a different question. While yes, a
    > (very) close-order convoy may have nearly the same probability of detection as a single ship, if
    > the U-boats then sank all of them once a convoy was detected, the average fraction of all ships
    > sunk would be the same either way.

    I don't believe this *ever* happened ("sank all at once"). Please correct me if I'm wrong. Even the
    probability of this happening seems amiss, what with submarines' limited weaponry, speed (especially
    near convoys) and defences.

    > If the U-boat was alone and only had time to sink one or two boats of the convoy, there would be
    > some protective effect. In reality multiple U-boats were steered to convoys once those were
    > detected, and wolf-packs formed (and their codes were broken, and the story gets more and more
    > complicated).

    "Steered to convoys" does not necessarily mean success in catching said convoys, even when they were
    detected (which was less and less frequent once Allied air-supremacy was attained). Also, U-boat
    losses were extrordinarily high, at up to a 35% chance of not returning to port once one put to sea
    near the war's end.

    > A major reason for convoy sailing was not probability of detection but the defense of the convoy,
    > as one escort vessel could protect multiple ships, and it would be impossible to escort all ships
    > if they were sailing individually.

    To quote a German Admiral at Kernevel in the Spring of 1941: "Our biggest headache is to *find* the
    enemy." --The Sea Wolves, Wolgang Frank, 1955, Holt, Reinhart and Winston Inc. This is not to
    disagree with your defence argument (a valid point), but rather to provide a contrast to your
    "reason for convoy sailing was not probability of detection" (which misses the point, as convoys
    didn't reduce the probability of detection, but allowed "nearly as good a chance of a convoy of
    forty ships in close order slipping unperceived", as one; and as agreed by you above, of course, if
    I understood you... "While yes, ..."). And I don't believe the absence of anything related to escort
    protection in Churchill's statement meant he didn't feel that this was a factor, rather I'd think it
    was just an obvious contributing factor not mentioned in this particular discussion on relative
    ocean/convoy sizes. Regards, Brett.
     
  7. Anon.

    Anon. Guest

    knowknot@mindspring.com wrote:
    > Would some statistically-knowledgeable folk be good enough to explain the mathematical/statistical
    > reasoning (or, if applicable, the principles which contrave) the following statement by Nobel
    > Physicist Steven Weinberg (writing about books on war, not about physics) in the most recent (Nov.
    > 6, 2003) issue of the "New York Review of Books"?
    >
    <snip>

    This question has now appeared on sci.stat.math, which is probably a better forum now as we're
    moving into the probabilistic side of the problem.

    I've forwarded Joe's reply there, as I think having some idea about the reality helps-. I would
    suggest people move, shoal-like, over to that forum if they want to remain on-topic.

    Bob

    --
    Bob O'Hara

    Rolf Nevanlinna Institute
    P.O. Box 4 (Yliopistonkatu 5) FIN-00014 University of Helsinki Finland Telephone: +358-9-191 23743
    Mobile: +358 50 599 0540 Fax: +358-9-191 22 779 WWW: http://www.RNI.Helsinki.FI/~boh/
     
  8. Brett Aubrey

    Brett Aubrey Guest

    "Anon." <bob.ohara@SOD.OFF.Spammers.helsinki.fi> wrote in message
    news:bo6s1l$1tkk$1@darwin.ediacara.org...
    > Frank Reichenbacher wrote:
    <snip >
    > > the difference in frequency of hits of the nail and the pin are both near zero and you said it's
    > > not true(?). Mine is a true statement.
    > What you wrote was: "The difference in the frequency of hits (probably near zero in both cases) is
    > insignificant." And I was pointing out that the difference was not insignificant.

    In the context of Churchill's statement ("The size of the sea is so vast ..."), Frank's initial
    point (agreement with Churchill) was correct, and his analogy was a basically valid, simple and
    useful illustration. Churchill was suggesting that the difference between ship and convoy sizes is
    *effectively irrelevant* when compared to the (very roughly) 6 to 8 million square mile area in
    which they sailed (of a total ~12M sq. mi.). Yes, there's a difference, but it "shinks in comparison
    almost to insignificance". I'll also note that the relative sizes in the ocean/convoy/ship vs. the
    dartboard/nail/pin analogy should mean the submarine would have far fewer sightings than the dart
    would have hits.

    I also find it interesting just how easy it was for U-boats to sink hundreds of ships just off the
    US coast in the "pre-convoy" days of the war, but how much harder it was for the Germans later on,
    even with a far larger submarine fleet (admittedly, there were a variety of factors at play here).
    Regards, Brett.
     
  9. Brett Aubrey

    Brett Aubrey Guest

    "Anon." <bob.ohara@SPAMMERS.SOD.OFF.helsinki.fi> wrote
    in message news:bobfte$6fl$1@darwin.ediacara.org...
    > Brett Aubrey wrote:
    > > "Anon." <bob.ohara@SOD.OFF.Spammers.helsinki.fi> wrote in message
    > > news:bo6s1l$1tkk$1@darwin.ediacara.org...
    > >>Frank Reichenbacher wrote:
    > > <snip >
    > >>>the difference in frequency of hits of the nail and the pin are both near zero and you said
    > >>>it's not true(?). Mine is a true statement.
    > >>What you wrote was: "The difference in the frequency of hits (probably near zero in both cases)
    > >>is insignificant." And I was pointing out that the difference was not insignificant.
    > > In the context of Churchill's statement ("The size of the sea is so so vast..."), Frank's
    > > initial point (agreement with Churchill) was correct, and his analogy was a basically valid,
    > > simple and useful illustration. Churchill was suggesting that the difference between ship and
    > > convoy sizes is *effectively irrelevant* when compared to the (very roughly) 6 to 8 million
    > > square mile area in which they , sailed (of a total ~12M sq. mi.). Yes there's a difference, but
    > > it "shinks in comparison almost to insignificance".
    >
    > No it doesn't, just because the ocean is large, the relative sizes of the one- and many- ship
    > fleets don't change. A 200m long boat is still twice as long as a 100m long boat no matter if it's
    > in the Atlantic Ocean or the Kensington Round Pond.
    >
    > Indeed, the fact that the ocean is large makes the whole thing clearer, because if you try to hit
    > a big boat in a small pond, it's too
    easy.

    Ahhh! Finally! Thanks for the explanation... Now I can see your source of confusion. Churchill was
    not discussing "hitting" anything at all! Nor was I and nor (I doubt) was Frank, except in his
    analogy. The discussion, rather, was around visual acquisition, not targeting once acquisition had
    been made. Anything relating to Churchill's vastness of the sea becomes completely irrelevant once
    contact has been made, unless it is subsequently lost again visually, which happened far too often
    for the Germans not to complain about it.

    That Churchill is talking about visual acquisition is clear in his phrase "slipping unperceived...".
    And I mentioned this in my initial response
    (10/29;12:55) with "the difficulties with visual acquisition", and later
    (11/29; 21:34) quoted a German Admiral to back up Churchill's claim: "Our biggest headache is to
    *find* the enemy." There was no talk around targeting or hitting anything *except* in the
    dartboard analogy. And an analogy it was - still a valid one, BTW; rather than a direct apples-to-
    apples comparison, at least in my opinion (Frank?).

    But Frank's analogy aside, Chuchill (and the German Admiral and I) were discussing sightings or lack
    thereof, while you're talking about "hitting a big boat" (presumably you mean ship) after it's been
    found - the two are not even remotely the same animal (and to reiterate the reasoning behind this,
    that "the size of the sea is so vast" is critical to the former, but irrelavant to the latter). If
    you still take issue with this, I look forward to your comments. Best regards, Brett.

    > And yes, I can throw some more formal maths at this (for those interested, look at the Poisson
    > approximation to the binomial distribution). What will make a difference is the way detectability
    > scales with size of fleet, but that is not the issue you were raising. Bob
    > --
    > Bob O'Hara
    >
    > Rolf Nevanlinna Institute
    > P.O. Box 4 (Yliopistonkatu 5) FIN-00014 University of Helsinki Finland Telephone: +358-9-191 23743
    > Mobile: +358 50 599 0540 Fax: +358-9-191 22 779 WWW: http://www.RNI.Helsinki.FI/~boh/
     
  10. Brett Aubrey

    Brett Aubrey Guest

    "Anon." <bob.ohara@SOD.OFF.Spammers.helsinki.fi> wrote in message
    news:bodsm0$u64$1@darwin.ediacara.org...
    > Brett Aubrey wrote:
    > > "Anon." <bob.ohara@SPAMMERS.SOD.OFF.helsinki.fi> wrote in message
    > > news:bobfte$6fl$1@darwin.ediacara.org...
    > >>Brett Aubrey wrote:
    > >>>In the context of Churchill's statement ("The size of the sea is so so vast..."), Frank's
    > >>>initial point (agreement with Churchill) was correct, and his analogy was a basically valid,
    > >>>simple and useful illustration. Churchill was suggesting that the difference between ship and
    > >>>convoy sizes is *effectively irrelevant* when compared to the (very roughly) 6 to 8 million
    > >>>square mile area in which they , sailed (of a total ~12M sq. mi.). Yes there's a difference,
    > >>>but it "shinks in comparison almost to insignificance".
    > >>No it doesn't, just because the ocean is large, the relative sizes of the one- and many- ship
    > >>fleets don't change. A 200m long boat is still twice as long as a 100m long boat no matter if
    > >>it's in the Atlantic Ocean or the Kensington Round Pond. Indeed, the fact that the ocean is
    > >>large makes the whole thing clearer, because if you try to hit a big boat in a small pond,
    > >>too easy.
    > > Ahhh! Finally! Thanks for the explanation... Now I can see your source of confusion. Churchill
    > > was not discussing "hitting" anything at all! Nor was I and nor (I doubt) was Frank, except in
    > > his analogy. The discussion, rather, was around visual acquisition, not targeting once
    > > acquisition had been made. Anything relating to Churchill's vastness of the sea becomes
    > > completely irrelevant once contact has been made, unless it is subsequently lost again visually,
    > > which happened far too often for the Germans not to complain about it.
    >
    > Visual acquisition is still the same problem - at the simplest level, you run around until you see
    > a boat. The larger the boat, the more likely you are to see it. It still doesn't matter if you're
    > running around in a pond or an ocean - you're still more likely to see several boats together than
    > to see one.

    Well *of course you're still more likely to see this*... Churchill said nothing to the contrary (and
    nor did I). In fact, he unambiguously implied this difference and I specifically mentioned this
    difference. But now you're implying that this was *your* idea, and that it diverged from
    Churchill's... Did you see Churchill as somehow stating: "The difference between the size of a
    convoy and the size of a single ship makes absolutely no difference whatsoever." or "There was in
    fact just as good a chance of a convoy of forty ships slipping unperceived between the patrolling
    U-boats as there was for a single ship."

    Similarly, did you miss my statement that "Yes, there's a difference"?

    The point, once again, is that the difference "shinks in comparison almost to insignificance" - Hey,
    we may not be that far away from agreement! Lemme try another scenario, hopefully closer to the
    topic than the dartboard analogy, though I still like that one even more than mine...

    *** Take a box of 24 ping-pong balls connected in a matrix by 2 inch strings, paint them
    blueish/green, and drop said matrix arbitrarily somewhere into the Pacific; then take another
    similarly camoflaged single ball and drop it arbitrarily somewhere into the Pacific. Hire 10 yachts
    to look for the balls/matrices. Repeat the ball drops up to several times a year for 4 years if so
    desired, to provide a bettter sample size for the "experiment". ... Now it would be my contention
    (or forecast in this case) that "the difference between the size of a matrix of balls and the size
    of the single ball is of such insignificance that the chance of the single ball being spotted is
    nearly as good as the chance of the matrix being spotted, before either reach landfall. ***

    Is there any agreement with this anology? Yes, there's a difference between the matrix and the ball,
    but that very small difference is effectively irrelevant given the sizes and numbers being
    discussed. And I will absolutely grant to you that the same experiment in different contexts would
    not yield the same results - for example a single ball vs. the matrix in the (smallish) lake by our
    cottege with 1,000 craft searching! (BTW, this also points to the criticality of the "vastness of
    the sea", which neither you nor Phillip mention much.) Could it be that this is a simple discussion
    of symantics? One would hope so. I'm still comfortable that Churchill's statement is valid, and that
    was the starting point. Comments? Best regards, Brett.

    > Bob
    > --
    > Bob O'Hara
    >
    > Rolf Nevanlinna Institute
    > P.O. Box 4 (Yliopistonkatu 5) FIN-00014 University of Helsinki Finland Telephone: +358-9-191 23743
    > Mobile: +358 50 599 0540 Fax: +358-9-191 22 779 WWW: http://www.RNI.Helsinki.FI/~boh/
     
  11. Brett Aubrey

    Brett Aubrey Guest

    "phillip smith" <deletethis-phills@ihug.co.nz> wrote in message
    news:bodslv$u4o$1@darwin.ediacara.org...
    > in article bocb6t$f7a$1@darwin.ediacara.org, Brett Aubrey at brett.aubrey@home.com wrote on
    > 6/11/03 3:21 PM:
    > > "Anon." <bob.ohara@SPAMMERS.SOD.OFF.helsinki.fi> wrote in message
    > > news:bobfte$6fl$1@darwin.ediacara.org...
    > >> Brett Aubrey wrote:
    > >>> "Anon." <bob.ohara@SOD.OFF.Spammers.helsinki.fi> wrote in message
    > >>> news:bo6s1l$1tkk$1@darwin.ediacara.org...
    > >>>> Frank Reichenbacher wrote:
    > >>> <snip >
    > >>>>> the difference in frequency of hits of the nail and the pin are both near zero and you said
    > >>>>> it's not true(?). Mine is a true statement.
    > >>>> What you wrote was: "The difference in the frequency of hits (probably near zero in both
    > >>>> cases) is insignificant." And I was pointing out that the difference was not insignificant.
    > >>> In the context of Churchill's statement ("The size of the sea is so so vast..."), Frank's
    > >>> initial point (agreement with Churchill) was correct, and his analogy was a basically valid,
    > >>> simple and useful illustration. Churchill was suggesting that the difference between ship and
    > >>> convoy sizes is *effectively irrelevant* when compared to the (very roughly) 6 to 8 million
    > >>> square mile area in which they , sailed (of a total ~12M sq. mi.). Yes there's a difference,
    > >>> but it "shinks in comparison almost to insignificance".
    > >> No it doesn't, just because the ocean is large, the relative sizes of the one- and many- ship
    > >> fleets don't change. A 200m long boat is still twice as long as a 100m long boat no matter if
    > >> it's in the Atlantic Ocean or the Kensington Round Pond. Indeed, the fact that the ocean is
    > >> large makes the whole thing clearer, because if you try to hit a big boat in a small pond, it's
    > >> too easy.
    > > Ahhh! Finally! Thanks for the explanation... Now I can see your source of confusion. Churchill
    > > was not discussing "hitting" anything at all! Nor was I and nor (I doubt) was Frank, except in
    > > his analogy. The discussion, rather, was around visual acquisition, not targeting once
    > > acquisition had been made. Anything relating to Churchill's vastness of the sea becomes
    > > completely irrelevant once contact has been made, unless it is subsequently lost again visually,
    > > which happened far too often for the Germans not to complain about it.
    >
    > The Relationship between convoy size and safety is not to do with the size of the target but the
    > relationship between size of the convoy and circumference.

    Well, I think this changes the initial topic, but I need clarification on your meaning to be sure.
    For now, though, I'll gladly try to respond. First, the initial discussion was between a specific
    form or "type" of safety (Churchill's "convoy ... slipping unperceived between the patrolling U-
    boats") and a "convoy size" of many (i.e. numerically only, with compactness as a given) and - if
    you choose to call it such - a "convoy size" of only one (1, uno) vs. the central and seemingly
    overlooked point of the vastness of the seas. Issues outside these - or for that matter issues than
    fail to take these into account - do not bear on the discussion, IMO. Not that I mind a change in
    direction as long as it's not clouding earlier points, as I think these are.

    I don't think I've stated that "the size of the target" is meaningful, so am unclear on your point
    negating this, above. In fact, this phrase can be interpreted in several ways, so I'm even unclear
    of your context. And safety (unqualified, as you have it above) will introduce many factors not
    even addressed by Churchill's quote or anything I've stated, so am again unclear on why you're
    bring this up. Please elaborate on your reason for mentioning these, and I will try to address them
    in more detail.

    The safety factor of "slipping unperceived" by the enemy affects the overall safety of the convoy,
    but so do many factors other than your "size of the convoy and circumference" (speed and
    manouverability of the least capable ships, numerical relationship of escorts to convoy ships,
    escort ship capabilities, intelligence, etc., etc.) More imoportantly, your "relationship between
    size of the convoy and circumference" was already stated in rough terms by Churchill's "in close
    order", so any variation in this relationship is outside the initial topic. If I'm missing what
    you're trying to get at, I apologise; again, please elaborate on your reasons for mentioning these,
    and I will try to address them in more detail

    > If I have ten convoys of of ten ships I have a greater circumference to area ration than if I have
    > one convoy of 100 ships. If I only put my anti submarine vessels around the circumference then the
    > ratio of anti submarine vessels per convoy member goes down with the increase in convoy sizes.
    > Therefore I can protect more vessels with fewer ships.

    No major dispute here, *if* you assume your eleven convoys are roughly equal in station-keeping,
    that is, typically, are *equally* "in close order". But in reality, your analogy is oversimplified -
    my remembering of 8 years in anti-submarine operations and on submarines (4 years each), is that
    while the convoy itself is in close order, the anti-submarine forces are neither equally in close
    order to the rest of the convoy nor are placed equally around the ships being protected. But of
    note, this again seems irrelevant to the initial discussion - there was nothing stated or even
    implied about protecting more (or less) vessels with fewer (or greater) ships in Churchill's quote.
    It was simply a single ship vs. a convoy. Please clarify if I'm missing something.

    > What has this got to do with evolution. Well there are two close relation ships CH Waddington who
    > discovered canalistation I'm pretty sure worked in operations research during the war and was
    > responsible along with many others for applying statistical methods to the problem of sinking
    sybmarines
    > and JBS haldane who gave us haldanes rule and haldanes delemma, spent his time during the, war
    > working on submarine escape systems which would have saved at least some german submariners lives,
    > had they had access to his work. Its tenuous I agree but its there.

    I presume you mean canalisation, but again, this seems outside of the discussion (i.e. not just
    tenous). As are submarine escape systems. But if you feel I'm wrong, please explain the relationship
    to me. Since we all know Churchill's quote but IMO digress from it on occasion, I'll attempt to
    reiterate in my own format - hopefully not that different from Churchill's - just what *is* germane
    to the discussion (please especially note Points 1 and 2, as I view these as critical to whether
    Churchill was correct or not, but see no mention of them from you):
    - The size of the sea, specifically its "vastness".
    - The huge difference between the vast seas *in comparison* to (tiny) convoys or single ships.
    - The far smaller difference between sizes of close-ordered convoys and single ships, to a point
    where it "shinks in comparison almost to insignificance".
    - The difficulties of visual acquistion for a submarine, as illustrated by Churchill's "slipping
    unperceived...".
    - The associated and illustrative fact that the chance of a convoy transiting undetected was nearly
    as good as for a single ship.

    Churchill and everyone else seem to agree that there is a difference in convoy size and ship size
    (what's there to find fault in that?), but where we seem to diverge is that this difference "shinks
    in comparison almost to insignificance when compared to the vast seas". If you agree with this
    analysis, is it fair to ask you to confine your comments to this divergence or the "bulleted" points
    immediately above, *just until* we can come to some sort of agreement or alternatively, agree to
    disagree? I suggest this because I think your bringing up issues like
    - "ten convoys of ten ships (vs.) one convoy of 100 ships",
    - "anti-submarine vessels around the circumference",
    - "the ratio of anti-submarine vessels per convoy member",
    - "the size of the target", etc., while all interesting and relevant to a discussion on convoys in
    general, bear absolutely no relation to the context of Churchill's quote, and more importantly, to
    the validity of Churchill's quote (which was the basis and the argument around this thread,
    respectively, IMO). Comments appreciated. Cheers! Best regards, Brett.

    > --
    > Phillip Smith phills@(buggger).co.nz replace bugger with ihug http://www.applied-evolution.co.nz
    >
    > "he who is smeared with blubber has the kindest heart" -- a Greenland Eskimo adage
     
  12. Anon.

    Anon. Guest

    Frank Reichenbacher wrote:
    > <knowknot@mindspring.com> wrote in message news:bnn5fs$ekl$1@darwin.ediacara.org...
    >
    <snip>
    >>(And, BTW, what might Weinberg have been referring to by his parenthetical reference to
    >>"many" fish?)
    >
    >
    > Churchill's characterization is correct. You can test it yourself if you have a dart board handy.
    >
    > Stick a 3-inch flathead nail on the dart board. Now try to hit it with a dart from the regulation
    > darting distance (4 meters?). Then replace the nail with a pin and try to hit it with a dart. The
    > difference in the frequency of hits (probably near zero in both cases) is insignificant.
    >
    Not true! In both cases the probability of hitting the nail/pin is small (especially the way I play
    darts!), but is (approximately) proportional to the area. You won't notice this unless you are
    patient enough to throw a lot of darts.

    Bob

    --
    Bob O'Hara

    Rolf Nevanlinna Institute
    P.O. Box 4 (Yliopistonkatu 5) FIN-00014 University of Helsinki Finland Telephone: +358-9-191 23743
    Mobile: +358 50 599 0540 Fax: +358-9-191 22 779 WWW: http://www.RNI.Helsinki.FI/~boh/
     
  13. Anon.

    Anon. Guest

    Brett Aubrey wrote:
    > "Anon." <bob.ohara@SOD.OFF.Spammers.helsinki.fi> wrote in message
    > news:bo6s1l$1tkk$1@darwin.ediacara.org...
    >
    >>Frank Reichenbacher wrote:
    >
    > <snip >
    >
    >>>the difference in frequency of hits of the nail and the pin are both near zero and you said it's
    >>>not true(?). Mine is a true statement.
    >>
    >>What you wrote was: "The difference in the frequency of hits (probably near zero in both cases) is
    >>insignificant." And I was pointing out that the difference was not insignificant.
    >
    >
    > In the context of Churchill's statement ("The size of the sea is so vast ..."), Frank's initial
    > point (agreement with Churchill) was correct, and his analogy was a basically valid, simple and
    > useful illustration. Churchill was suggesting that the difference between ship and convoy sizes is
    > *effectively irrelevant* when compared to the (very roughly) 6 to 8 million square mile area in
    > which they sailed (of a total ~12M sq. mi.). Yes, there's a difference, but it "shinks in
    > comparison almost to insignificance".

    No it doesn't, just because the ocean is large, the relative sizes of the one- and many- ship fleets
    don't change. A 200m long boat is still twice as long as a 100m long boat no matter if it's in the
    Atlantic Ocean or the Kensington Round Pond.

    Indeed, the fact that the ocean is large makes the whole thing clearer, because if you try to hit a
    big boat in a small pond, it's too easy. And yes, I can throw some more formal maths at this (for
    those interested, look at the Poisson approximation to the binomial distribution).

    What will make a difference is the way detectability scales with size of fleet, but that is not the
    issue you were raising.

    Bob

    --
    Bob O'Hara

    Rolf Nevanlinna Institute
    P.O. Box 4 (Yliopistonkatu 5) FIN-00014 University of Helsinki Finland Telephone: +358-9-191 23743
    Mobile: +358 50 599 0540 Fax: +358-9-191 22 779 WWW: http://www.RNI.Helsinki.FI/~boh/
     
  14. in article bocb6t$f7a$1@darwin.ediacara.org, Brett Aubrey at
    brett.aubrey@home.com wrote on 6/11/03 3:21 PM:

    > "Anon." <bob.ohara@SPAMMERS.SOD.OFF.helsinki.fi> wrote in message
    > news:bobfte$6fl$1@darwin.ediacara.org...
    >> Brett Aubrey wrote:
    >>> "Anon." <bob.ohara@SOD.OFF.Spammers.helsinki.fi> wrote in message
    >>> news:bo6s1l$1tkk$1@darwin.ediacara.org...
    >>>> Frank Reichenbacher wrote:
    >>> <snip >
    >>>>> the difference in frequency of hits of the nail and the pin are both near zero and you said
    >>>>> it's not true(?). Mine is a true statement.
    >>>> What you wrote was: "The difference in the frequency of hits (probably near zero in both cases)
    >>>> is insignificant." And I was pointing out that the difference was not insignificant.
    >>> In the context of Churchill's statement ("The size of the sea is so so vast..."), Frank's
    >>> initial point (agreement with Churchill) was correct, and his analogy was a basically valid,
    >>> simple and useful illustration. Churchill was suggesting that the difference between ship and
    >>> convoy sizes is *effectively irrelevant* when compared to the (very roughly) 6 to 8 million
    >>> square mile area in which they , sailed (of a total ~12M sq. mi.). Yes there's a difference, but
    >>> it "shinks in comparison almost to insignificance".
    >>
    >> No it doesn't, just because the ocean is large, the relative sizes of the one- and many- ship
    >> fleets don't change. A 200m long boat is still twice as long as a 100m long boat no matter if
    >> it's in the Atlantic Ocean or the Kensington Round Pond.
    >>
    >> Indeed, the fact that the ocean is large makes the whole thing clearer, because if you try to hit
    >> a big boat in a small pond, it's too
    > easy.
    >
    > Ahhh! Finally! Thanks for the explanation... Now I can see your source of confusion. Churchill was
    > not discussing "hitting" anything at all! Nor was I and nor (I doubt) was Frank, except in his
    > analogy. The discussion, rather, was around visual acquisition, not targeting once acquisition had
    > been made. Anything relating to Churchill's vastness of the sea becomes completely irrelevant once
    > contact has been made, unless it is subsequently lost again visually, which happened far too often
    > for the Germans not to complain about it.

    The Relationship between convoy size and safety is not to do with the size of the target but the
    relationship between size of the convoy and circumference If I have ten convoys of of ten ships I
    have a greater circumference to area ration than if I have one convoy of 100 ships. If I only put my
    anti submarine vessels around the circumference then the ratio of anti submarine vessels per convoy
    member goes down with the increase in convoy sizes. Therefore I can protect more vessels with fewer
    ships. What has this got to do with evolution. Well there are two close relation ships CH Waddington
    who discovered canalistation I'm pretty sure worked in operations research during the war and was
    responsible along with many others for applying statistical methods to the problem of sinking
    sybmarines and JBS haldane who gave us haldanes rule and haldanes delemma, spent his time during
    the, war working on submarine escape systems which would have saved at least some german submariners
    lives, had they had access to his work. Its tenuous I agree but its there.

    --

    Phillip Smith phills@(buggger).co.nz replace bugger with ihug http://www.applied-evolution.co.nz

    "he who is smeared with blubber has the kindest heart" -- a Greenland Eskimo adage
     
  15. Anon.

    Anon. Guest

    Brett Aubrey wrote:
    > "Anon." <bob.ohara@SPAMMERS.SOD.OFF.helsinki.fi> wrote in message
    > news:bobfte$6fl$1@darwin.ediacara.org...
    >
    >>Brett Aubrey wrote:
    >>
    >>>"Anon." <bob.ohara@SOD.OFF.Spammers.helsinki.fi> wrote in message
    >>>news:bo6s1l$1tkk$1@darwin.ediacara.org...
    >>>
    >>>>Frank Reichenbacher wrote:
    >>>
    >>><snip >
    >>>
    >>>>>the difference in frequency of hits of the nail and the pin are both near zero and you said
    >>>>>it's not true(?). Mine is a true statement.
    >>>>
    >>>>What you wrote was: "The difference in the frequency of hits (probably near zero in both cases)
    >>>>is insignificant." And I was pointing out that the difference was not insignificant.
    >>>
    >>>In the context of Churchill's statement ("The size of the sea is so so vast..."), Frank's initial
    >>>point (agreement with Churchill) was correct, and his analogy was a basically valid, simple and
    >>>useful illustration. Churchill was suggesting that the difference between ship and convoy sizes
    >>>is *effectively irrelevant* when compared to the (very roughly) 6 to 8 million square mile area
    >>>in which they , sailed (of a total ~12M sq. mi.). Yes there's a difference, but it "shinks in
    >>>comparison almost to insignificance".
    >>
    >>No it doesn't, just because the ocean is large, the relative sizes of the one- and many- ship
    >>fleets don't change. A 200m long boat is still twice as long as a 100m long boat no matter if it's
    >>in the Atlantic Ocean or the Kensington Round Pond.
    >>
    >>Indeed, the fact that the ocean is large makes the whole thing clearer, because if you try to hit
    >>a big boat in a small pond, it's too
    >
    > easy.
    >
    > Ahhh! Finally! Thanks for the explanation... Now I can see your source of confusion. Churchill was
    > not discussing "hitting" anything at all! Nor was I and nor (I doubt) was Frank, except in his
    > analogy. The discussion, rather, was around visual acquisition, not targeting once acquisition had
    > been made. Anything relating to Churchill's vastness of the sea becomes completely irrelevant once
    > contact has been made, unless it is subsequently lost again visually, which happened far too often
    > for the Germans not to complain about it.

    Visual acquisition is still the same problem - at the simplest level, you run around until you see a
    boat. The larger the boat, the more likely you are to see it. It still doesn't matter if you're
    running around in a pond or an ocean - you're still more likely to see several boats together than
    to see one.

    Bob

    --
    Bob O'Hara

    Rolf Nevanlinna Institute
    P.O. Box 4 (Yliopistonkatu 5) FIN-00014 University of Helsinki Finland Telephone: +358-9-191 23743
    Mobile: +358 50 599 0540 Fax: +358-9-191 22 779 WWW: http://www.RNI.Helsinki.FI/~boh/
     
  16. "Anon." <bob.ohara@SOD.OFF.Spammers.helsinki.fi> wrote in message
    news:bnsd06$1suv$1@darwin.ediacara.org...
    > Frank Reichenbacher wrote:
    > > <knowknot@mindspring.com> wrote in message news:bnn5fs$ekl$1@darwin.ediacara.org...
    > >
    > <snip>
    > >>(And, BTW, what might Weinberg have been referring to by his parenthetical reference to "many"
    > >>fish?)
    > >
    > >
    > > Churchill's characterization is correct. You can test it yourself if you have a dart board
    > > handy.
    > >
    > > Stick a 3-inch flathead nail on the dart board. Now try to hit it with a dart from the
    > > regulation darting distance (4 meters?). Then replace the
    nail
    > > with a pin and try to hit it with a dart. The difference in the
    frequency
    > > of hits (probably near zero in both cases) is insignificant.
    > >
    > Not true!

    Yes it is!

    In both cases the probability of hitting the nail/pin is
    > small (especially the way I play darts!),

    That's what I said!

    but is (approximately)
    > proportional to the area. You won't notice this unless you are patient enough to throw a lot
    > of darts.

    Yeah! I know!

    I don't think you are getting this. It's really pretty simple. Maybe you just have something against
    Churchill, or maybe Weinberg, that's getting in your way.

    I said the difference in frequency of hits of the nail and the pin are both near zero and you said
    it's not true(?). Mine is a true statement. You could stand there all day tossing darts at the board
    with either a pin or a nail stuck on it and you might hit the nail once or twice but you almost
    certainly wouldn't hit the pin. I'm guessing, therefore, that out of 1000 attempts each you might
    hit the nail twice and the pin not at all. Therefore, the probability of hitting the nail is .002,
    while the probability of hitting the pin is zero. Both are near enough zero that the difference is
    insignificant.

    In the WWII example, the pin is analagous to a single boat, while the nail is analagous to a convoy,
    and the dart is analagous to a U-Boat. The average accuracy of a dart throw is analagous to a U-
    Boat's ability to detect enemy ships in the ocean. In the fish example, the pin is analagous to a
    single fish, the nail is analagous to a school of fish, and the dart is analagous to a predator. The
    average accuracy of a dart throw is analagous to a predator's ability to sense prey.

    The probability of hits is proportional to the relative difference in size of the pins, the nails,
    and the dart tips combined with the average accuracy of a throw. If that's what you mean by "area",
    then you are correct and we agree.

    Frank

    >
    > Bob
    >
    > --
    > Bob O'Hara
    >
    > Rolf Nevanlinna Institute
    > P.O. Box 4 (Yliopistonkatu 5) FIN-00014 University of Helsinki Finland Telephone: +358-9-191 23743
    > Mobile: +358 50 599 0540 Fax: +358-9-191 22 779 WWW: http://www.RNI.Helsinki.FI/~boh/
     
  17. "Anon." <bob.ohara@SOD.OFF.Spammers.helsinki.fi> wrote in
    news:bnsd06$1suv$1@darwin.ediacara.org:

    > Frank Reichenbacher wrote:
    >> <knowknot@mindspring.com> wrote in message news:bnn5fs$ekl$1@darwin.ediacara.org...
    >>
    > <snip>
    >>>(And, BTW, what might Weinberg have been referring to by his parenthetical reference to
    >>>"many" fish?)
    >>
    >>
    >> Churchill's characterization is correct. You can test it yourself if you have a dart board handy.
    >>
    >> Stick a 3-inch flathead nail on the dart board. Now try to hit it with a dart from the regulation
    >> darting distance (4 meters?). Then replace the nail with a pin and try to hit it with a dart. The
    >> difference in the frequency of hits (probably near zero in both cases) is insignificant.
    >>
    > Not true! In both cases the probability of hitting the nail/pin is small (especially the way I
    > play darts!), but is (approximately) proportional to the area. You won't notice this unless you
    > are patient enough to throw a lot of darts.
    >
    > Bob
    >

    You are correct about the specific question described in the dartboard analogy, but Churchill's
    characterization about the chance of detecting a convoy is still correct - and here's the reason why
    (this puzzled me for a bit, but I think I'm right):

    In the dart situation you have a point intersection with a two- dimensional target. In the convoy
    situation you have a two-dimensional intersection (the circle defined by the distance at which a sub
    can visually acquire a target) with a two dimensional target. The better analogy would be to
    throwing suction cup tipped darts at a dartboard with 40 nails either scattered around the board or
    concentrated together at one location, and determining the odds of any part of the suction cup
    hitting a nail. As Joe Felsenstein has pointed out, if the entire group of nails is wiped out once
    one is detected then the overall odds of ship survival is the same, but if a majority of the ships
    in the convoy will survive a sub encounter then the convoy is the better choice (regardless of the
    protection factor).

    And yes this should go to sci.stat.math, but it isn't on my list and the chances of it being added
    are slim and none :). Feel free to forward this. But just to make this relevant to evolution (and
    thus not tick off our esteemed moderator), it would seem that this strategy for the most part would
    not help fish (or other schooler/herders), since they in general are not moving from one safe haven
    to another while subject to predation in between. Once acquired as a target by the predator, the
    predator can simply follow the herd/school and continue to pick them off without suffering any
    increased risk.

    Yours,

    Bill Morse
     
  18. Anon.

    Anon. Guest

    Frank Reichenbacher wrote:
    > "Anon." <bob.ohara@SOD.OFF.Spammers.helsinki.fi> wrote in message
    > news:bnsd06$1suv$1@darwin.ediacara.org...
    >
    >>Frank Reichenbacher wrote:
    >>
    >>><knowknot@mindspring.com> wrote in message news:bnn5fs$ekl$1@darwin.ediacara.org...
    >>>
    >>
    >><snip>
    >>
    >>>>(And, BTW, what might Weinberg have been referring to by his parenthetical reference to "many"
    >>>>fish?)
    >>>
    >>>
    >>>Churchill's characterization is correct. You can test it yourself if you have a dart board handy.
    >>>
    >>>Stick a 3-inch flathead nail on the dart board. Now try to hit it with a dart from the regulation
    >>>darting distance (4 meters?). Then replace the
    >>
    > nail
    >
    >>>with a pin and try to hit it with a dart. The difference in the
    >>
    > frequency
    >
    >>>of hits (probably near zero in both cases) is insignificant.
    >>>
    >>
    >>Not true!
    >
    >
    > Yes it is!
    >
    >
    > In both cases the probability of hitting the nail/pin is
    >
    >>small (especially the way I play darts!),
    >
    >
    > That's what I said!
    >
    >
    > but is (approximately)
    >
    >>proportional to the area. You won't notice this unless you are patient enough to throw a lot
    >>of darts.
    >
    >
    > Yeah! I know!
    >
    > I don't think you are getting this. It's really pretty simple. Maybe you just have something
    > against Churchill, or maybe Weinberg, that's getting in your way.
    >
    > I said the difference in frequency of hits of the nail and the pin are both near zero and you said
    > it's not true(?). Mine is a true statement.

    What you wrote was: "The difference in the frequency of hits (probably near zero in both cases) is
    insignificant."

    And I was pointing out that the difference was not insignificant.

    You could
    > stand there all day tossing darts at the board with either a pin or a nail stuck on it and you
    > might hit the nail once or twice but you almost certainly wouldn't hit the pin.

    Right, so you agree with me that there is a significant difference - you'll hit the nail many more
    times than you'll hit the pin.

    <snip>
    > The probability of hits is proportional to the relative difference in size of the pins, the nails,
    > and the dart tips combined with the average accuracy of a throw. If that's what you mean by
    > "area", then you are correct and we agree.
    >
    It's actually approximately proportional to the area (you're assuming that over the area of the
    nail, the probability distribution for where the dart strikes is uniform - I would make the same
    assumption, but as the nail head gets bigger, this assumption gets worse)

    Bob

    --
    Bob O'Hara

    Rolf Nevanlinna Institute
    P.O. Box 4 (Yliopistonkatu 5) FIN-00014 University of Helsinki Finland Telephone: +358-9-191 23743
    Mobile: +358 50 599 0540 Fax: +358-9-191 22 779 WWW: http://www.RNI.Helsinki.FI/~boh/
     

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