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Hi thickhead ,
I don't mean to modify your formula to mine .
I mean perhaps I can generalize my formula for P(2)
to P(3) in my form , and I already have some ideas .
Solid geometry seems cannot be applied any more .
I will try by following the pattern of P(2) .
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I actually wanted to find whether one can be derived from the other. in my formula i cannot use a or b directly if they are > 1/2 but this restriction is not there in yours.
{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 ,
Will you state more and give some examples ?
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Results
a b m n mr.wong thickhead
0.6 0.1 0.1 0.4 0.065740741 0.065740741
0.6 0.2 0.2 0.4 0.141666667 0.141666667
0.6 0.3 0.3 0.4 0.225 0.225
0.6 0.4 0.4 0.4 0.311111111 0.311111111
0.6 0.5 0.4 0.5 0.393333333 0.393333333
0.6 0.6 0.4 0.4 0.466666667 0.466666667
0.6 0.7 0.3 0.4 0.525 0.525
0.6 0.8 0.2 0.4 0.566666667 0.566666667
0.6 0.9 0.1 0.4 0.591666667 0.591666667
0.6 0.95 0.05 0.4 0.597916667 0.597916667
0.6 0.99 0.01 0.4 0.599916667 0.599916667Formulas transposed for better view
a 0.6
b 0.1
m =MIN(K63,K64,1-K63,1-K64)
n =MAX(MIN(K63,1-K63),MIN(K64,1-K64))
mr.wong =MIN(K63,K64)-(MIN(1-K63,1-K64)^3-MAX(0,1-K63-K64)^3)/(3*(1-K63)*(1-K64))
thickhead =(K65*K66*(1-K66)-K65^3/3)/(1-K63)/(1-K64)Last edited by thickhead (2016-07-01 18:36:37)
{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 aim I re-wrote your original formula to the one
in # 47 is to eliminate the restriction of the order of
the original m and n ( i.e. K65 and K66 ) defined by
you . This is always the benefit of using functions of
min and max though it looks a bit too prolix . I think
there is no need to emphasize too much in this point .
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I am interested in how to bring co relation between the two formula for the same phenomenon logically. It is strange I do not have 1-a-b in my formula at all. it seems converting to m and n bypassed it. but i want the logic. Previously I had checked only some values but now I have compared for all value of ranging from 0.1 to 0.99. I only wonder how you found your formula without logical steps to achieve it.
{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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Last edited by thickhead (2016-07-03 21:08:01)
{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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Last edited by thickhead (2016-07-19 00:33:38)
{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 ,
Once you have assumed a < b , then a = min (a,b) and
1-b = min ( 1-a,1-b ) ; thus formula (3) can be re-written
as : min(a,b)- min( 1-a,1-b) ^3 - max(0,1-a-b) ^3/ 3(1-a)(1-b) .
If you have assumed a > b , the formula will remain the
same since then b= min(a,b) and 1-a = min ( 1-a,1-b) .
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Yes. if I have to remove the tag a<b I have to write that way but the way I have written is less repulsive because it contains only one max function getting rid of 2 min() functions and one can always start with a<b. In fact I have only proved your formula and achieved nothing new.
{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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Last edited by thickhead (2016-07-06 21:24:30)
{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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3 segments
a b c l m n abc formula lmn formula
0.2 0.3 0.4 0.2 0.3 0.4 0.036706349 0.036706349
0.6 0.7 0.8 0.2 0.3 0.4 0.513888889 0.513888889
0.1 0.6 0.8 0.1 0.2 0.4 0.062731481 0.062731481
0.5 0.5 0.5 0.5 0.5 0.5 0.25 0.25
0.3 0.8 0.9 0.1 0.2 0.3 0.279761905 0.279761905
0.3 0.7 0.9 0.1 0.3 0.3 0.256349206 0.256349206
0.15 0.6 0.9 0.1 0.15 0.4 0.102083333 0.102083333{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 ,
I need time to analyse your formula for P(3) .
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Hi thickhead ,
Please check your formula in #61 carefully .
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I have checked it carefully. Please look at he table in #62
{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 ,
For a = 1/4 , b = 1/3 and c = 1/2 , what will be the results for
both formulas ?
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a b c l m n abc formula lmn formula
0.25 0.333333333 0.5 0.25 0.333333333 0.5 0.068383488 0.068383488{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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Thanks thickhead ,
The formula should be valid . But unless it can be simplified ,
in practical use the lmn formula will be much faster .
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{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 ,
Since I cannot derive a formula for P(3) readily , I have
tried to solve problem involving 3 moving segments with
P(2) step by step . For example , in problem with segment E
being 1 unit long and 3 moving segments A , B and C each being
1/2 unit , I found the expected length of common portion of A and B firstly .
The formula P(2) gives the common portion of A and B ( denoted
by D , should also be considered as a segment .) to be 1/3 un . Then
I substituted D and C into P(2) again , but I got a wrong answer other than
the correct answer .( the expected length of common portion of A and B and C
should be 1/4 un . and the corresponding probability = 1/4 .)
This may be because the segment D so obtained will no longer be
movable . It is fixed at certain position(s) in E . Thus P(2) should
not be applied again . Instead , another formula involving 1 movable
segment and 1 fixed segment with appointed position should be used .
Can you derive such a formula ?
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Hi thickhead ,
Perhaps we may do it in another way .
As we got the intermediate product D , the common
portion of A and B being 1/3 unit , can we adjust
its length to a value k so that after substituted it to
the original formula P(2) with C being 1/2 un . we can
get the final correct result 1/4 .
Can we find the rule for such adjustment ?
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k=0.38345 works only for segment sizes of 1/2.Also all 3 1/2 segments can be replaced by 2 segments of size 0.43584 each. but on what basis?
{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 ,
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 .
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{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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For segments of different lengths it is more complicated. if l,m,n are numbers associated with the lengths as before viz min(a,1-a) etc arranged in ascending order, and a,b,c are the corresponding lengths we can build a rule for 3 segments from the result of 2 segments. Keep l for the third segment.
{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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