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#76 2016-07-19 20:52:15

thickhead
Member
Registered: 2016-04-16
Posts: 1,086

Re: geometric probability--segments

mr.wong wrote:

Hi  thickhead ,

We  have  found  that  for  3  variables  , if  a = 1/2 ,
b =1/3  and  c = 1/4 , then  P = 709/10368  by  P(3)
directly .
If  we  treat  a  and  b  firstly  with  P(2)  , we  get  an 
intermediate  value  d = 23/108 , if  we  want  to  find 
the  value  of  corresponding  k , perhaps  we  have  to 
solve  the  equation :
min(k,1/4) - [min ( 1-k, 3/4)^3 - max(0, 3/4- k)^3 ] / 9/4*(1-k)
= 709/10368 
In  fact  I  really  don't  know  how  to  solve  it  !
So  it  seems  this  is  not  a  feasible  way   to  find  P(3)
by  P(2)  indirectly .

value of k comes to 0.23258.


{1}Vasudhaiva Kutumakam.{The whole Universe is a family.}
(2)Yatra naaryasthu poojyanthe Ramanthe tatra Devataha
{Gods rejoice at those places where ladies are respected.}

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#77 2016-07-20 02:07:02

thickhead
Member
Registered: 2016-04-16
Posts: 1,086

Re: geometric probability--segments

Regarding some property,
if a=1/2,b=1/3 and c=3/4 P=3*709/10368.
if a=1/2,b=2/3 and c=1/4 P=2*709/10368.
if a=1/2,b=2/3 and c=3/4 P=2*3*709/10368.
we can call it property of conjugates.


{1}Vasudhaiva Kutumakam.{The whole Universe is a family.}
(2)Yatra naaryasthu poojyanthe Ramanthe tatra Devataha
{Gods rejoice at those places where ladies are respected.}

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#78 2016-07-21 16:21:20

mr.wong
Member
Registered: 2015-12-01
Posts: 252

Re: geometric probability--segments

Hi  thickhead ,

It  is  quite  interesting  for  your  property  of  conjugates .
But  it  seems   only  valid  if  the  original  variables  are 
1/2 , 1/3  and  1/4 . Also  if  c  is  changed  from  1/4  to 
2/4 ( = 1/2) , P  will  not  be  double . ( I  got  P = 25/ 162 )
Can  this  property  be  generalized ?

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#79 2016-07-21 17:16:22

thickhead
Member
Registered: 2016-04-16
Posts: 1,086

Re: geometric probability--segments

The property relates only to conjugates.This is because in my formula of 'lmn", numerator consists of only l,m,and n the minor conjugates. if a=l and it changes to a=1-l,the numerator remains same but the denominator part relating to a changes from 1-a to [1-(1-a)] and it is easy to incorporate this change.


{1}Vasudhaiva Kutumakam.{The whole Universe is a family.}
(2)Yatra naaryasthu poojyanthe Ramanthe tatra Devataha
{Gods rejoice at those places where ladies are respected.}

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#80 2016-07-23 16:26:21

mr.wong
Member
Registered: 2015-12-01
Posts: 252

Re: geometric probability--segments

Hi  thickhead ,

The  property  of  conjugates  can  also  be  applied  to  problems 
involving  2  variables . For  example :  P ( 1/3 , 1/4 ) = 29/ 288 ,
while  P( 1/3 , 1- 1/4) = P( 1/3 , 3/4 ) = 29/ 96  = 3* 29/ 288 ;
similarly  P( 1-1/3, 1/4) = P( 2/3, 1/4) = 29/144 = 2* 29/288 ;
and  P( 1-1/3 , 1-1/4 ) = P( 2/3, 3/4 ) = 29/ 48 = 6* 29/288  .

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#81 2016-07-23 16:33:22

thickhead
Member
Registered: 2016-04-16
Posts: 1,086

Re: geometric probability--segments

Actually I had detected it first with 2 moving segments only.


{1}Vasudhaiva Kutumakam.{The whole Universe is a family.}
(2)Yatra naaryasthu poojyanthe Ramanthe tatra Devataha
{Gods rejoice at those places where ladies are respected.}

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#82 2016-07-24 23:42:18

mr.wong
Member
Registered: 2015-12-01
Posts: 252

Re: geometric probability--segments

This  property  also  applies  for  variables  with  numerator 
other  than  1 . For  example :  P( 1/3 , 2/5 ) = 137/810 ,  while 
P( 1/3 , 3/5 ) = 137/ 540  =  3/2 * 137/810   .
In  general  , for  0 ≤ a ≤ 1  and  0 ≤ b ≤ 1 ,
this  formula  may  be  true :  P( 1-a , b) = (1-a / a) * P( a , b) .
Thus  P(1-a,1-b) = [(1-a)(1-b)/ab] * P(a,b)

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