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    Re: That darned old cocked hat
    From: Antoine Couëtte
    Date: 2010 Dec 10, 06:17 -0800

    Hello to all,

    ... with a special attention in particular to John Karl, George Huxtable and Peter Fogg.

    PRELIMINARY NOTE : Sorry for all our non "math addicts" Members here. No Worries Mates !! CelNav is MUCH more interesting and fun that this on-going debate.

    BIG PICTURE : It looks like we are loosing / have lost track of a very interesting debate between George and John about the probability level of the actual fix being inside the 3 side cocked hat.

    ******
    ******

    John,

    in " [NavList 14687] That darned old cocked hat " you enclosed a very interesting study to be found here : http://www.fer3.com/arc/imgx/Darned-Old-Cocked-Hat.pdf in which you stated in particular :

    QUOTE

    2. Second, the probability that the ship is outside of the cocked hat is 84%.

    UNQUOTE

    *******

    George,

    You strongly questioned this assertion through a reasoning totally different from John's. And you have indicated that - whatever the size of the cocked hat - such probability is 75% only outside the hat, with a full 25% probability inside the hat.

    *******

    Peter,

    On your side you have questioned and opposed George's reasoning, because in your view if the cocked hat were reduced to just one single point - which may very well happen since absolutely nothing forbids it - allocating a 25% probability to just this unique point makes no sense at all for you.

    I would take position here only on one point Peter, namely your remark itself mentionned here-above. While I am sometimes under the impression that math might happen to be taught differently on each side of the Atlantic Ocean now and then :-) , I can see absolutely nothing which forbids to allocate a full 25% probability onto just one single point. So, I do not think that your view point - or at least what I have understood about it - is a counter example to George's results.

    *******

    John, to you again,

    what is interesting to me here is that you are using a "statistics reasoning method" which can produce remarkable results, including the ones you just recently published in our thread "A simple three-body fix puzzle" . I should add that not only Frank, but also other people - Kermit included - find your approach very clever and are impressed.

    So, on this specific "darned old cocket hat", and after the last detailed explanation from George, I would feel that the ball is your courtyard, John, through either defeating George's arguments in a fully convincing manner, or through "fine tuning" your own use of the statistical method you you have run so brightly just a few days ago.

    I would be highly interested when your both results agree. For sure there is a lot to be learnt through the use of your probability method. For the time being, and until you have not reached agreement together with George, I still feel quite hungry or thirsty.

    And one last note : Please John, be so kind as to tell me how you interpret the ellipses "tag numbers" in your .pdf document. They are still unclear to me. For example, for the most inner ellipse wearing a "0.013 tag" just around the Least Square Distances Point, what does the "0.013 value" exactly mean ? I cannot reasonably reconcile such (apparently very low) value with any kind of reasonable area probability level. Thanks again to you John for your expected help here.

    *******
    *******

    Finding the truth (25% ? or less ? or more ? inside the cocked hat)- and not at all finding who is right / wrong here - is the challenge to our NavList Community.

    Additionnaly, I have found partial information elsewhere which tends to support George's results. (See the Post Sriptum down below).

    I also think that this question has already been settled in the past by various authoritative people, and possibly including some of us on this NavList forum, in addition to George's previous posts on the matter.

    *******
    *******

    Best Regards to all NavList Members, and in particular to you John, George and Peter,

    and many thanks for your Kind Attention, and upcoming replies.


    Antoine


    Antoine M. "Kermit" Cou�tte

    *******

    PS : In Volume XXXI of the French NAVIGATION Journal, Issue #121 dtd Jan 1983, on page 45 all the way down, we can read the following note which I just translated to French to English :

    QUOTE

    DANIELS mentions a fascinating point explained in "Admiralty Navigation Manual (1947)" where it is proven that [in the case of 3 observations] the probability for the actual fix to lay inside the cocked hat is only 0.25 . Such probability increases to 0.5 inside the inner area surrounded by 4 LOP's, and it reaches only 0.965 for the inner area surrounded by 9 LOP's. The general applicable formula is as follows :

    p (n) = 1 - [ 1 - n / { 2 ** (n-1)}]

    UNQUOTE
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