*** Probability of Area Derivation ***

To:           Search & Rescue Info <SARINFO@mindlink.bc.ca>
Date sent:    Mon, 07 Aug 95 10:58:17 -0700
From:         jim_clark@om.cv.hp.com
Subject:      Derivation of simplification.
To:           sarinfo@mindlink.bc.ca
Item Subject: cc:Mail Text

Dear Martin Colwell,

Here is an electronic copy of the derivation I did in class. The equations
as they were being taught in class were cumbersome and the instructor
struggled with the math. The class was getting sleepy and I flipped ahead
through the notes.
Looking ahead on page 215 in the MSO manual (1990 (third) edition)
were some simplified equations that the teacher didn't use, under "New
Formula, Old Idea". I decided to determine if they were the same equations
only simplified. I quickly proved to myself that they were the same; so,
during break I wrote the following up as overheads and presented them to
the class after break.

The following is an over explained version of what I presented.


  Pa =  old POA for segment A before segment A was searched
  Pb =  old POA for segment B before segment A was searched
  Pa' = new POA for segment A after segment A was searched
  Pb' = new POA for segment B after segment A was searched
  PaD = POD for segment A that was just searched

  These are the equations as they were being taught:

           Pa(1-PaD)
  Pa' = ----------------
        Pa(1-PaD)+(1-Pa)

        Pb(1-Pa')
  Pb' = ---------
         (1-Pa)



  First I expanded Pa':

          Pa(1-PaD)
  Pa' = -------------
        Pa-PaPaD+1-Pa

  the Pa-Pa cancel in the bottom and simplifies to

        Pa(1-PaD)
  Pa' = ---------

where PaPaD = old POA of segment A times the POD of segment A just searched.



  Second I expanded Pb':
  knowing that

       (6-2)
  (1 + -----)
         4

  is equal to


  (4 + (6-2))
  -----------
       4

  I expand Pb'

        Pb(1-Pa')
  Pb' = ---------
         (1-Pa)

  to

               (Pa(1-PaD)
        Pb(1 - ----------)
                (1-PaPaD)
  Pb' = ------------------
              (1-Pa)

  which becomes


           Pb((1-PaPaD)-Pa(1-PaD))
           -----------------------
                 (1-PaPaD)
  Pb' = -----------------------------
                   (1-Pa)


  and knowing that

        8
        -
        4
      -----
        2

  equals

        8
       ---
       4x2

  we get

        Pb((1-PaPaD)-Pa(1-PaD))
  Pb' = -----------------------
           (1-PaPaD)(1-Pa)

  which expands to


        Pb(1-PaPaD-Pa+PaPaD)
  Pb' = --------------------
          (1-PaPaD)(1-Pa)

  the PaPaD-PaPaD cancel in the top and simplifies to


           Pb(1-Pa)
  Pb' = ---------------
        (1-PaPaD)(1-Pa)

  the (1-Pa) in the top and bottom cancel and the result is

           Pb
  Pb' = ---------
        (1-PaPaD)


  Therefore the two equations as they were taught simplify from/to




           Pa(1-PaD)       Pa(1-PaD)
  Pa' = ---------------- = ---------
        Pa(1-PaD)+(1-Pa)   (1-PaPaD)

  and

        Pb(1-Pa')      Pb
  Pb' = --------- = ---------
         (1-Pa)     (1-PaPaD)

  since POA times POD = the probability of success I call
  (1 - POA POD) the probability of non-success which refers to (1-PaPaD).


  It was interesting that as I was sitting down I got a rousing round of
  applause. I think the applause was because I was sitting down rather than
  because of the information I presented.


  I hope this helps.


  James E. Clark
  (503) 967-9188
  4771 Bramblewood Lane N.W.
  Albany, OR, USA
  97321-9372

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