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#26 2016-01-18 15:28:36

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

Re: geometric probability--segments

Hi  bobbym ,

For  example  , let  a = b = 1/10 , then  your  formula  will  get

P = ( 1/100 - 2/10 +  3/100 - 3/10 + 1 ) / ( - 27/10 )

    = [( 1 +3 +100 - 20 - 30 ) / 100 ] / ( -27/10 )

    =  ( 54/100 ) *( - 10 / 27 )

    =  - 1/ 5

Another  example  , let  a = b = 2/5 ,

my  formula  yields   P = 28/135
while  your  formula  yields  P = 1/5 = 27/135 ;
which  one  is  more  correct ?

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#27 2016-01-18 17:32:19

bobbym
bumpkin
From: Bumpkinland
Registered: 2009-04-12
Posts: 109,606

Re: geometric probability--segments

Hi;

The formula I posted was for

when a>b:

when a = b


In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

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#28 2016-01-20 14:14:51

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

Re: geometric probability--segments

Hi  bobbym ,

Your  second  formula  is  too  complicated  , I  can't   even
check  its  validity  for  a = b =  2/5 .
Don't  spend  time  on  your  formula  anymore !  Just  use
my  formula  for  further  investigation  if  you  have  checked  its  validity .

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#29 2016-01-20 14:52:04

Relentless
Member
Registered: 2015-12-15
Posts: 631

Re: geometric probability--segments

Hi;

I do not know what this thread is about, but the bottom equation can be simplified if you define x=a=b:

Which is 28/135 when x=a=b is 2/5

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#30 2016-01-20 17:11:59

bobbym
bumpkin
From: Bumpkinland
Registered: 2009-04-12
Posts: 109,606

Re: geometric probability--segments

Hi;

Don't  spend  time  on  your  formula  anymore !  Just  use
my  formula  for  further  investigation  if  you  have  checked  its  validity .

If you do not want it then there is no point but the reason I favor replacing non analytical functions like min and max with algebraic expressions is because they are much easier to derive more information from. For instance, relentless' simplification. Also we can integrate and differentiate algebraic expressions to derive means and standard deviation and other moments, this will be difficult with min and max.


In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

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#31 2016-01-22 15:48:01

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

Re: geometric probability--segments

Hi  Relentless ,

Thanks  for  your  reply !  Though  your  formula  is  valid
for  x=a=b=2/5 , but  if  you  substitute  x=a=b  by  3/5   ,
you  will  get  a  not  so  correct  result  9/20  instead  of
7/15 .

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#32 2016-01-22 15:51:51

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

Re: geometric probability--segments

Hi  bobbym ,

The  use  of  function  with  min  and  max  will  save  much labours , for  it  applies  to  every  possible  conditions  but  not  just  certain  conditions .
I  pay  respect  to  your  favour  and  hard  works , but    time  and  spirit  should  be  spent  on  the  right  place .
Later  I  shall  discuss  some  related  topics  in  a  new  thread .

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#33 2016-01-22 16:08:52

Relentless
Member
Registered: 2015-12-15
Posts: 631

Re: geometric probability--segments

Hi mr wong, bobbym's formula also gives 9/20 for a=b=3/5. I have confirmed graphically that my formula and his are equivalent - if the answer should be 7/15 then both formulas are definitely wrong. In fact, no positive number will give an answer of 7/15 ... I am getting a=b~-0.869 as the only solution for 7/15 (the exact form is complicated).

But again, I am afraid I do not know what we are talking about! Haha

Last edited by Relentless (2016-01-22 16:11:59)

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#34 2016-01-22 17:27:07

bobbym
bumpkin
From: Bumpkinland
Registered: 2009-04-12
Posts: 109,606

Re: geometric probability--segments

I agree that 7 / 15 is correct. The formulas I posted are only valid for 0 < a < 1/2, a >= b. Putting 3 / 5 in them violates that condition.


In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

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#35 2016-06-22 05:34:07

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

Re: geometric probability--segments


I got 28/135 for  a=b=3/5 as well as 7/15 for 2/5. I shall bring it out later.


{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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#36 2016-06-22 20:22:26

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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#37 2016-06-22 23:02:26

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

Re: geometric probability--segments

Hi  thickhead ,

Should  it  be  28/135 for a=b=2/5  as  well  as 
7/15  for  3/5 ?  ( # 35 )

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#38 2016-06-22 23:39:55

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

Re: geometric probability--segments

Last edited by thickhead (2016-06-23 04:37:41)


{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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#39 2016-06-23 16:54:09

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

Re: geometric probability--segments

Hi  thickhead ,

It  seems  you  have  developed  another  valid  formula 
for  2  moving  segments  which  looks  quite  different 
with  mine , though  it  appears  a  bit  too  complicated .
Moreover , your  formula  seems  not  symmetric  for 
a  and  b .  More  data  should  be  necessary  for  proving 
its  validity  . ( I  suggest  you  take  reference  to  my  formula .)
However  , after   your  formula  has  been  confirmed
I  hope  you  can  modify  it  to  one  applicable  for  3  segments ( variables ) .

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#40 2016-06-23 17:45:43

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

Re: geometric probability--segments

Last edited by thickhead (2016-06-23 21:29:04)


{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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#41 2016-06-25 00:15:57

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

Re: geometric probability--segments

bobbym wrote:

Hi;

The first bit of real results in a long time of hard work.

If we denote the length of the three segments that will slide along E, S1,S2 and S3.

1) Then:

That covers some of the cases, I have more but need time to check them...

This equation is surprisingly synonymous with what I derived. bobbym is very mysterious. Never discloses how he derived.


{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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#42 2016-06-25 03:37:06

bobbym
bumpkin
From: Bumpkinland
Registered: 2009-04-12
Posts: 109,606

Re: geometric probability--segments

No mystery, just a set of rules.

1) When posting to a student on a homework question, I will sometimes give a method but not the answer and more times the answer without a method. We believe that a problem solver must be able to back engineer a method from the solution. This is a very valuable skill. The student has to do at least part of the work if he is to learn anything.

2) Sometimes the OP just is looking for an answer. With that answer, mr.wong just wanted to verify his own work so a solution was not necessary.

3) It is easy in my opinion for someone doing EM to get the answer. Whereas in classical mathematics they thrash around with algebraic symbols until an answer finally shows itself, in EM we often have the answer but not the method. That is why rule 1 is so important.

4) This is not a rule but a fact of life, sometimes I can easily get the answer but am not smart enough to develop a method that will satisfy a math type.


In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

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#43 2016-06-25 04:30:23

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

Re: geometric probability--segments

Bobbym
Thank you for clarification.


{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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#44 2016-06-25 21:57:39

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

Re: geometric probability--segments

Last edited by thickhead (2016-06-27 01:45:02)


{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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#45 2016-06-26 04:12:21

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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#46 2016-06-26 04:36:30

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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#47 2016-06-26 21:46:11

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

Re: geometric probability--segments

Hi  thickhead ,

It  is  quite  astonished  that  for  problem  involving 
2  segments  , both  formula  which  looks  quite  different 
are  in  fact  identical .  ( In  # 44, the  term  (1-c)  should  be
removed  from  my  formula ! )

For  my  own  convenience  I  have  re-written  your  formula 
for  2  variables  in  the  following .( Where 
m  denotes  min [min (a,1-a),min (b,1-b) ]  and 
n  denotes   max [min (a,1-a),min (b,1-b)]  ,  thus  there 
is  no  need  to  consider  the  order  of  m  and  n . )

P(2) = [mn(1-n)- m^3 /3 ] / (1-a)(1-b)
  =  min ( a, 1-a, b, 1-b )*  max [min (a,1-a),min (b,1-b)]
* { 1- max [min (a,1-a),min (b,1-b)]}
-   min ( a, 1-a, b, 1-b ) ^3 / 3   /  (1-a) (1-b)

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#48 2016-06-26 21:53:00

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

Re: geometric probability--segments

Hi mr.wong,
I would like the basis on which your formula was derived. Is it in existence for a long time ? I have started the subject from scratch and had no exposure to it earlier.


{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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#49 2016-06-27 23:06:13

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

Re: geometric probability--segments

Hi  thickhead ,

In  fact  I  developed  the  formula  more  than  20  years 
ago !  I  haven't  publish  it  to  the  public  until  last  year 
in  a  Chinese  math . forum . I  got  the  formula  through 
solid  geometry  and  had  not  expected  that  the  related
problem  can  also  be  solved  with  integration   and  derive 
a  formula  which  looks  quite  different  with  mine  ! I  hope
my  formula  can  be  modified  and  generalized  to  one  involving  3  variables .

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#50 2016-06-27 23:53:03

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

Re: geometric probability--segments

Hi mr.wong,
You said my formula can be modfied to yours but how do you explain max(0,1-a-b) term ?
1-a-b is the gap left over if A and B are joined end to end. and min(1-a,1-b) is the gap left if A and B are superposed.
but I am unable to interpretation of those terms in the formula. One more thing where these ideas can be implemented in practice?


{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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