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mr.wong

Member

Registered: 2015-12-01

Posts: 252

Re: Probability --- triangles

Hi  thickhead ,

Can  you  confirm  your  answer   ( 1/10 )  of  E involving  2  moving  triangles 
by  getting  the  result  directly  with  multiple  integration  other  than  indirectly  deduced  from  results  of  the  4  smaller  triangles  of  E ?

thickhead

Member

Registered: 2016-04-16

Posts: 1,086

Re: Probability --- triangles

The local probability function ( depending on the position (x,y) of the point ) differs for the 4 smaller triangles. I have to evaluate them separately and combine.

---

{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.}

mr.wong

Member

Registered: 2015-12-01

Posts: 252

Re: Probability --- triangles

Hi  thickhead ,

Will  you  expect  apart  from  isosceles  right  angle  triangles ,
the  results  will  remain  the  same  for  any  other  triangles , say 
equilateral  triangles ?

bobbym

bumpkin

From: Bumpkinland

Registered: 2009-04-12

Posts: 109,606

Re: Probability --- triangles

There is a demonstration for any type of triangle as long as all your conditions apply.

---

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.

mr.wong

Member

Registered: 2015-12-01

Posts: 252

Re: Probability --- triangles

Hi bobbym ,

If  the  theory  of  that  demonstration  is  too  hard  for  me ,
then  I  shall  skip  it .
In  the  following  we  shall  try  to  deduce  some  related  results  from 
what  we  have  got .
With  reference  to  the  original  post  in  this  thread :
Let  P ( A | E ) denotes  P  of  the  point  chosen in E  which  also  lies  in  A .
and  P ( ∼A | E )  denotes  P  of  the  point  chosen in E  which  lies  outside  A
being 1- 1/4 = 3/4 .
(1) Let  P ( A ∩ B | E ) denotes  P  of  the  point  lies  inside  both  A  and  B  [ the  intersection  of  A  and  B ] , with  value  being  1/10  ( according  to  thickhead 's  result ) , thus  P ( ∼A ∩ B | E ) = P ( B | E ) - P ( A ∩ B | E ) = 1/4 - 1/10 = 3/20 . being  P  of  the  point  lies  outside  A  but  inside  B .
( Notice  that  it  should  not  be  calculated  as   P ( ∼A | E ) * P ( B | E ) = 
3/4 * 1/4 = 3/16 .)
(2) Since  P ( A ∩ B ∩ C | E ) = 1/21  ( thickhead's  result ) , thus 
P ( ∼A ∩ B ∩ C | E ) =  P ( B ∩ C | E ) - P ( A ∩ B ∩ C | E )
=  1/10 - 1/21 = 11/210   being  P  of  the  point  lies  outside  A  but  inside 
both  B  and  C  .
(3) P (∼ A ∩∼ B ∩ C | E ) = P ( C | E ) - P ( A ∩∼ B ∩ C | E ) -
  P ( A ∩ B ∩ C | E ) - P ( ~ A ∩ B ∩ C | E ) = 1/4 - 11/210 - 1/21 - 11/210
=  41/ 420   (  may  be  illustrated  more  clearly  with  a  Venn  Diagram )
Related  problem (I) :
Find  the  probability  of  a  point  chosen  in  E  randomly  which  lies  within 
one  and  only  one  moving  triangle . ( answer : 41/ 140 )

bobbym

bumpkin

From: Bumpkinland

Registered: 2009-04-12

Posts: 109,606

Re: Probability --- triangles

Will  you  expect  apart  from  isosceles  right  angle  triangles ,
the  results  will  remain  the  same  for  any  other  triangles , say
equilateral  triangles ?

It at least shows that this is true provided his work is correct.

---

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.

thickhead

Member

Registered: 2016-04-16

Posts: 1,086

Re: Probability --- triangles

For 2 moving triangles in E

Last edited by thickhead (2016-06-17 03:37:46)

---

{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.}

mr.wong

Member

Registered: 2015-12-01

Posts: 252

Re: Probability --- triangles

Thanks  thickhead ,

Elegant  and  laborious  work !
Since  the  common  portion  of  2  similar  and  parallel  triangles 
is  still  a  similar  and  parallel  triangle ( This  is  not  true  for  squares ) , 
originally  I  had  expected  that  the  answer  should  be  the  square  of  certain 
average  ratio  of  the  corresponding  sides  of  overlapped  triangle  with  E .
In  fact   it  is  much  more  complicated .

thickhead

Member

Registered: 2016-04-16

Posts: 1,086

Re: Probability --- triangles

Once the probability functions have been  derived you can even get for 10 moving triangles just by using these integrations  but with power of 10 but it is not possible for "n" moving triangles as the integrals are not that easy for general terms. In case of squares it was possible for n squares. here also the troublesome is the integration in triangle X, the others can be managed.

---

{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.}

mr.wong

Member

Registered: 2015-12-01

Posts: 252

Re: Probability --- triangles

Related  problem (II) :
If  a  point  is  chosen  randomly  in  square  LQMN , find  the  probability 
that  the  point  lies  inside  at  least  2  moving  triangles .
( answer :  32/105 )

thickhead

Member

Registered: 2016-04-16

Posts: 1,086

Re: Probability --- triangles

Hi bobbym ,

If  the  theory  of  that  demonstration  is  too  hard  for  me ,
then  I  shall  skip  it .
In  the  following  we  shall  try  to  deduce  some  related  results  from 
what  we  have  got .
With  reference  to  the  original  post  in  this  thread :
Let  P ( A | E ) denotes  P  of  the  point  chosen in E  which  also  lies  in  A .
and  P ( ∼A | E )  denotes  P  of  the  point  chosen in E  which  lies  outside  A
being 1- 1/4 = 3/4 .
(1) Let  P ( A ∩ B | E ) denotes  P  of  the  point  lies  inside  both  A  and  B  [ the  intersection  of  A  and  B ] , with  value  being  1/10  ( according  to  thickhead 's  result ) , thus  P ( ∼A ∩ B | E ) = P ( B | E ) - P ( A ∩ B | E ) = 1/4 - 1/10 = 3/20 . being  P  of  the  point  lies  outside  A  but  inside  B .
( Notice  that  it  should  not  be  calculated  as   P ( ∼A | E ) * P ( B | E ) = 
3/4 * 1/4 = 3/16 .)
(2) Since  P ( A ∩ B ∩ C | E ) = 1/21  ( thickhead's  result ) , thus 
P ( ∼A ∩ B ∩ C | E ) =  P ( B ∩ C | E ) - P ( A ∩ B ∩ C | E )
=  1/10 - 1/21 = 11/210   being  P  of  the  point  lies  outside  A  but  inside 
both  B  and  C  .
(3) P (∼ A ∩∼ B ∩ C | E ) = P ( C | E ) - P ( A ∩∼ B ∩ C | E ) -
  P ( A ∩ B ∩ C | E ) - P ( ~ A ∩ B ∩ C | E ) = 1/4 - 11/210 - 1/21 - 11/210
=  41/ 420   (  may  be  illustrated  more  clearly  with  a  Venn  Diagram )
Related  problem (I) :
Find  the  probability  of  a  point  chosen  in  E  randomly  which  lies  within 
one  and  only  one  moving  triangle . ( answer : 41/ 140 )

Is this your answer or required correct answer?
Alternate way:
Probability of point lying in A but not B &C=

Corrected to avoid confusion later.

Last edited by thickhead (2016-06-20 16:43:44)

---

{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.}

mr.wong

Member

Registered: 2015-12-01

Posts: 252

Re: Probability --- triangles

Hi  thickhead ,

You  got  the  same  answer  of  41/420  with  me .
(where  the  term  2/20  should  be  2/10 ) But  sometimes 
it  may  cause  confusion   since  P  stands  for  1/4  ( thus 
( 1-P )  should be  3/4 )  while  in  the  expression 
P-2P^2 + P^3  , P^2  stands  for  P  for  2  moving 
triangles ( instead  of  1/4 * 1/4 =1/16 ) and  P^3  stands  for  P  for  3  moving  triangles  .( instead  of  1/4* 1/4 * 1/4 =1/64 ) .
But  I  wonder  why  it  seems  you  can  always  get  a  correct  answer  by  your  way ! ( including  P  for  triangle  A  or  triangle  X )

thickhead

Member

Registered: 2016-04-16

Posts: 1,086

Re: Probability --- triangles

Last edited by thickhead (2016-06-20 16:48:26)

---

{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.}

thickhead

Member

Registered: 2016-04-16

Posts: 1,086

Re: Probability --- triangles

Related  problem (II) :
If  a  point  is  chosen  randomly  in  square  LQMN , find  the  probability 
that  the  point  lies  inside  at  least  2  moving  triangles .
( answer :  32/105 )

I presume there are 3 moving triangles. i am getting answer as 213/2688. is your answer reliable?

---

{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.}

thickhead

Member

Registered: 2016-04-16

Posts: 1,086

Re: Probability --- triangles

---

{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.}

mr.wong

Member

Registered: 2015-12-01

Posts: 252

Re: Probability --- triangles

Hi  thickhead ,

For  square  LQMN  I  mean  the  square  taken  from 
triangle  P (0,2) ,Q (0,0) R (2,0) , thus  its  coordinates 
will  be  L ( 0,1) , Q (0,0) ,M ( 1,0) and N (1,1) . So  a  diagram  is  always  helpful !
Should  L  be  (0,2) in  your  problem ?

thickhead

Member

Registered: 2016-04-16

Posts: 1,086

Re: Probability --- triangles

I thought triangle E is doubled and you renamed it LQMN. Yes L (0,2) I had taken. Not only that I had to derive 8 probability functions separately. may be useful in future.

Last edited by thickhead (2016-06-21 16:53:42)

---

{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.}

thickhead

Member

Registered: 2016-04-16

Posts: 1,086

Re: Probability --- triangles

---

{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.}

mr.wong

Member

Registered: 2015-12-01

Posts: 252

Re: Probability --- triangles

Related  problem :

Let E denotes a triangle PQR with PQ = QR = 6 units  and
angle Q = 90 degree . Inside  E  there  are  2  similar  triangles 
A  and  B  both  being  parallel  with  E  and  can move freely and 
uniformly  inside  E but  must keep parallel with E in moving .
The  lengths  of  the  corresponding  sides  of  A  and  B  are 
3  units  and  2  units  respectively .
If  a  point  is  chosen  randomly  on E , find  the  probability  that
the  point  also  lies  inside  A  and  B  at  the  same  time .

mr.wong

Member

Registered: 2015-12-01

Posts: 252

Re: Probability --- triangles

Hi  thickhead ,

Will  the  property  of conjugates  also  applies  to  triangles ? 
For  example , if  the  length  of  the  corresponding  side  of 
B  is  changed  to  4  units , will  the  corresponding  probability  be  double ?
or  something  like  that ?

thickhead

Member

Registered: 2016-04-16

Posts: 1,086

Re: Probability --- triangles

In triangles so far we have dealt with moving triangles whose sides are 1/2 of stationary triangle. Before attempting your problem of #44 I want to try the problem with triangles A ,B  and Cwith sides=2 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.}

thickhead

Member

Registered: 2016-04-16

Posts: 1,086

Re: Probability --- triangles

An interesting point with triangle E with vertices (0,0) (1,0)and (1,0) and moving triangle A of side a is that
(1) For a<1/3 there is a central triangle of sides 1-3a (with running track of width

just inside the borders of E) which will have constant local probability  a^2/(1-a^2).
(2)For a>2/3 there will be central triangle of sides 3a-2 (with running track of width

just inside the borders of E) which will have constant local probability  1 , since it is always within triangle A.

Last edited by thickhead (2016-08-02 16:21:37)

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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.}

mr.wong

Member

Registered: 2015-12-01

Posts: 252

Re: Probability --- triangles

Hi  thickhead ,

For  (1)  should  it  be  a < 1/3 ?

thickhead

Member

Registered: 2016-04-16

Posts: 1,086

Re: Probability --- triangles

Hi mr.wong,
You are right. I corrected it.

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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.}