"Anon."  wrote in
news:bnsd06$1suv$1{at}darwin.ediacara.org: 

> Frank Reichenbacher wrote:
>>  wrote in message
>> news:bnn5fs$ekl$1{at}darwin.ediacara.org...
>> 
> 
>>>(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
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