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#51 2016-06-28 15:53:11

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

Re: geometric probability--segments

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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#52 2016-06-28 16:28:42

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

Re: geometric probability--segments

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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#53 2016-07-01 15:01:54

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

Re: geometric probability--segments

Hi  thickhead ,

Will  you  state  more  and  give  some  examples ?

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#54 2016-07-01 16:42:14

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

Re: geometric probability--segments

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	0.599916667

Formulas 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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#55 2016-07-02 16:31:35

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

Re: geometric probability--segments

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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#56 2016-07-02 17:10:30

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

Re: geometric probability--segments

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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#57 2016-07-03 20:28:18

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

Re: geometric probability--segments

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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#58 2016-07-03 20:31:53

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

Re: geometric probability--segments

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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#59 2016-07-03 23:00:57

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

Re: geometric probability--segments

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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#60 2016-07-03 23:48:32

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

Re: geometric probability--segments

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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#61 2016-07-06 21:07:20

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

Re: geometric probability--segments

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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#62 2016-07-07 00:26:10

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

Re: geometric probability--segments

 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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#63 2016-07-07 15:50:45

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

Re: geometric probability--segments

Hi  thickhead ,

I  need  time  to  analyse  your  formula  for  P(3) .

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#64 2016-07-07 22:57:08

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

Re: geometric probability--segments

Hi  thickhead ,

Please  check  your  formula  in  #61  carefully .

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#65 2016-07-08 05:35:35

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

Re: geometric probability--segments

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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#66 2016-07-08 15:10:34

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

Re: geometric probability--segments

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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#67 2016-07-08 15:41:07

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

Re: geometric probability--segments

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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#68 2016-07-08 22:55:21

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

Re: geometric probability--segments

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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#69 2016-07-09 00:10:24

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

Re: geometric probability--segments


{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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#70 2016-07-09 21:28:57

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

Re: geometric probability--segments

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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#71 2016-07-15 16:11:32

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

Re: geometric probability--segments

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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#72 2016-07-18 19:39:15

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

Re: geometric probability--segments

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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#73 2016-07-18 20:41:30

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

Re: geometric probability--segments

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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#74 2016-07-18 21:09:52

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

Re: geometric probability--segments


{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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#75 2016-07-19 04:54:50

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

Re: geometric probability--segments

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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