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What I currently find intriguing is the enlarged hat which retains the same
proportions as the original. I suspect that it does not need to be enlarged greatly
to increase enormously the likelihood of its containing the actual position, as
soon as we accept the notion that the further we move from the LOPs the less the
chance of encountering the actual position.

The next step which is, inevitably, finding the fix position in exactly the same
place as it was in the smaller hat of the same proportions, doesn't just beg but
fairly shrieks the question, now thrice repeated: is this not all an exercise in
futility?  25%, 14.whatever%, what difference does it make if the fix position, the
only point we can calculate and use, is exactly the same?

George Huxtable wrote:

> I suspect he is gradually coming round to the majority way of thinking
> about the cocked-hat, but desperately trying to avoid admitting it, even to
> himself. He is doing this by redefining the quantities involved. We can't
> allow him to get away with it.

I would like to think that I retain an open mind on the question. Whether it is the
'majority way' or not is irrelevant, of course, this doesn't affect whether it is
right or wrong.  As for 'redefining', another term might be lateral thinking.

Still suspect that the base for the 25% idea has a slim and tottery foundation: the
50% chance of a RANDOM point falling on either side of a RANDOM line, which becomes
25% when two lines intersect.
As somebody else pointed out, they are not random at all, each LOP is an
approximation of the actual position, this is why they come together in a
(hopefully) small hat. This is what I meant with my woefully unscientific
expression of the opposing LOPs 'pulling' the probability of the AP toward
themselves.

'I took that to imply that he thought (then) that it was impossible for the
"real position" to be outside the cocked hat. Does he think that now?'

Never said so, thought so even less. See 'open mind' above.

'But this is in the world of Alice through the Looking Glass'

Lewis Caroll, interestingly enough, was a mathematician (and a few other quirky
things). No doubt his fertile imagination was useful to him in the pursuit of all
his interests.

Let's look at this (yet!) another way. Visually. Draw (with your mind's eye if you
are lazy) a hat with generous proportions within and without on an A4 sheet (as I
have been). Now according to the 25% theory put 25 points at random within the hat.
What are we to do with the 75 we have left to distribute? I suppose we have to
place them at the same random distances outside the hat. Some of them, since there
are so many compared to inside the hat, are going to land an awfully long way from
the hat itself. Nevertheless each is presumably as valid as any other. But no other
distribution, either, seems to make much sense. According to the 'LOP approximation
of AP' fact, and also the 'careful navigator' assumption which accompanied the
beginning of this discussion, since we still have only 25 for inside the hat the
remaining 75 will have to be thickly clustered all along, and close to, the outside
of the LOPs, since the further away from the LOPs we go the fewer points are to be
found. This picture is truly absurd, it looks as though the 75 outside are beating
at the walls, trying to get inside!

'We can't allow him to get away with it.'

The 25% theory is starting to sound like dogma, something that MUST be accepted.

We seem dangerously close to the 25% theory becoming a sacred cow itself.
Where is the butcher when we need him?

   

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