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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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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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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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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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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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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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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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