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#1 2016-05-30 15:34:48

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

Probability --- triangles

Let E denotes a triangle PQR with PQ = QR  and
angle Q = 90 degree . L , M and N are mid-points of PQ , QR and RP respectively . Let A denotes triangle PLN , B denotes triangle LQM , C denotes triangle NMR and X  denotes triangle LMN .
A , B and C can move freely and  uniformly  inside  E but  must keep parallel with E in moving .
(I)  If  a  point  is  chosen  randomly  on  triangle  PLN , find  the 
probability  that  the  point  also  lies  inside  C .
(II) If  a  point  is  chosen  randomly  on  triangle  LQM ,
find  the  probability  that  the  point  also  lies  inside  C .
Should  the  answers  of  (I)  and  (II)  be  the  same ?

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#2 2016-06-06 14:58:37

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

Re: Probability --- triangles

Assumed  that  nomatter  triangle  PLN , or  triangle  LQM , or 
triangle  NMR , their  probability  with  C  (also  for  A  and  B )
will  be  the  same  and  denoted  by  x .
Since  the  probability  of  E  with  C  should  simply  be  1/4  and 
the    probability  of  X  with  C  can  be  found  to  be  1/2 , thus 
1/4  =  1/4 * x + 1/4 * x + 1/4 * x + 1/4 * 1/2
       =  3/4 * x +  1/8 
Thus  1/8 = 3/4 * x   , which  yields  x = 1/8* 4/3 = 1/6 .

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#3 2016-06-06 17:35:01

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

Re: Probability --- triangles

Hi;

(I)  If  a  point  is  chosen  randomly  on  triangle  PLN , find  the
probability  that  the  point  also  lies  inside  C .

You are asking if (PLN)=A and C slide around what is the chance the point is in A and C?


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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#4 2016-06-06 21:10:26

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

Re: Probability --- triangles

Hi  bobbym ,

For  triangle  PLN   I  mean  that  when  triangle  A  is  fixed  at  its original   position , otherwise  it  will  be  stated  as    A  if  moved .

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#5 2016-06-07 00:39:02

thickhead
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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.}

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#6 2016-06-07 17:17:28

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

Re: Probability --- triangles

Last edited by thickhead (2016-06-07 18:10:51)


{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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#7 2016-06-07 22:24:53

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

Re: Probability --- triangles

Hi  thickhead ,

Thanks  much  for  your  work ! 
Thus  it  seems  my  assumption  in  # 2  should  be  correct , 
and  it  makes  the  further  study  much  simpler .
I  am  not  satisfied  for  the  questions  involving  only  1 
moving  triangle . The  following  related  questions  based  on 
the  original  problem  will  be  involving   2  moving  triangles .
Please  try  to  solve  them  with  your  skilful  technique  in
multiple  integration  if  you  are  interested !

(III) (a) If  a  point  is  chosen  randomly  on  triangle  PLN , find  the  probability  that  the  point  also  lies  inside  B  and  C .
(III) (b) If a point is chosen randomly on triangle X , find the
probability that the point also lies inside both B and C .
(III) (c) If a point is chosen randomly on triangle E , find the
probability that the point also lies inside B and C .

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#8 2016-06-09 21:24:21

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

Re: Probability --- triangles

Last edited by thickhead (2016-06-09 22:13:38)


{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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#9 2016-06-09 21:37:04

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

Re: Probability --- triangles

Hi  thickhead ,

Have  you  any  idea  to  solve  the  3  questions  of  problem  (III)  ?

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#10 2016-06-09 22:20:20

thickhead
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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.}

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#11 2016-06-09 22:45:41

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.}

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#12 2016-06-09 23:51:08

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

Re: Probability --- triangles

Thanks  thickhead  again ,

Triangle LQN  should  be LQM .
How  about  Problem (III) (c) ?

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#13 2016-06-10 18:55:19

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.}

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#14 2016-06-10 19:05:02

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

Re: Probability --- triangles

Hi wong,
please check all double integrals with"Wolfram alpha double integrals" available online with the result in proper fractions also.
Now for a point in E to be in 2 moving triangles we have to take weighted average of all 4 triangles.
P for E=

Last edited by thickhead (2016-06-10 19:05:35)


{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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#15 2016-06-11 15:40:32

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

Re: Probability --- triangles

Thanks  thickhead ,

I  even  don't  know  how  to  check  the  integrals 
with"Wolfram alpha double integrals" !

Hi  bobbym ,

Your  simulation  of  P = 1/10  (in  the  thread  "
geometric  probability  --- square " # 11  and  #  20 
has  been  verified  by  thickhead 's  work  , while 
my  results  were  all  wrong !

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#16 2016-06-11 17:00:28

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

Re: Probability --- triangles

Hi;

Could you please be a bit more specific as to what problem you mean. I do not wish to misunderstand you because this is important.


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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#17 2016-06-11 20:04:09

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

Re: Probability --- triangles

Hi  bobbym ,

The  related  problem  was  :

Inside  a  triangle  E  there  are  2  smaller  similar  triangles
A  and  B , both  with  length  of  relative  sides  being  1/2  of
that  of  E . All  the  3  triangles  are  parallel  with  vertices  upwards .A  and  B  can  move  freely  inside  E , but  must  keep  parallel  with  E .If  a  point  is  chosen  randomly  on  E ,
find  the  probability  that  the  point  lies  inside  A  and  B  at  the  same  time .

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#18 2016-06-11 20:54:36

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

Re: Probability --- triangles

Hi;

Then we all have a problem because the exact answer to that question is not 1 / 10. The exact analytical answer as given to me is 13 / 120.


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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#19 2016-06-12 05:06:40

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

Re: Probability --- triangles

I think this is the time for introspection. i had goofed up in my initial attempts but later I found logical approach.
Take a point T(x,y) and slide all the edges of the movable triangle along T. Trace out the exterior of the triangle. This gives a diagram of mobility of triangle when T is inside it, within infinite region. i do not what to call it but let us say "ABSOLUTE FAVORABLE MOBILITY DIAGRAM" We still have not defined the restrictions on the triangle. If we say it has to move in a triangle whose sides are double depending on where we choose t inside it, there will be intersection of the bigger triangle with ABSOLUTE FAVORABLE MOBILITY DIAGRAM which I may call GROSS  FAVORABLE MOBILITY DIAGRAM. if we place the moving triangle say PLN with one point say P touching the border of GROSS  FAVORABLE MOBILITY DIAGRAM from inside and move it all along the border and trace the path of any one particular point of PLN say N we get NET  FAVORABLE MOBILITY DIAGRAM which gives the area of mobilty of PLN for which T is inside it. Also if we move triangle PLN inside the triangle E in a similar fashion we get NET TOTAL MOBILITY DIAGRAM. The ratio of the two gives us probability of T being inside PLN. This probability is a function of x and y which is to be integrated over the area over which the probability expression is valid. if we have to find the average probability of being inside 2 moving triangles we have to take

the square of local probabilty p(x,y) and integrate over the required area and not use

Last edited by thickhead (2016-06-12 05:13:21)


{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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#20 2016-06-13 14:49:07

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

Re: Probability --- triangles

Hi  thickhead ,

Then  what  should  be  the  results ?

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#21 2016-06-13 15:01:41

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

Re: Probability --- triangles

Inside  a  triangle  E  there  are  2  smaller  similar  triangles
A  and  B , both  with  length  of  relative  sides  being  1/2  of
that  of  E . All  the  3  triangles  are  parallel  with  vertices  upwards .A  and  B  can  move  freely  inside  E , but  must  keep  parallel  with  E .If  a  point  is  chosen  randomly  on  E ,
find  the  probability  that  the  point  lies  inside  A  and  B  at  the  same  time .

The answer to this is 13 / 120, this is close to 1 / 10 which is what I got as an estimate of that probability. The guys at the SE would be glad to show you why it is 13 / 120 for the exact answer.


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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#22 2016-06-13 16:18:37

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

Re: Probability --- triangles

1/10 for a point chosen inside E to lie inside 2 moving triangles, 1/21 for 3 moving triangles.
It is 31/210 for a point chosen inside X to lie inside 3 moving 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.}

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#23 2016-06-13 18:56:21

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

Re: Probability --- triangles

bobbym wrote:

Inside  a  triangle  E  there  are  2  smaller  similar  triangles
A  and  B , both  with  length  of  relative  sides  being  1/2  of
that  of  E . All  the  3  triangles  are  parallel  with  vertices  upwards .A  and  B  can  move  freely  inside  E , but  must  keep  parallel  with  E .If  a  point  is  chosen  randomly  on  E ,
find  the  probability  that  the  point  lies  inside  A  and  B  at  the  same  time .

The answer to this is 13 / 120, this is close to 1 / 10 which is what I got as an estimate of that probability. The guys at the SE would be glad to show you why it is 13 / 120 for the exact answer.


Hi  bobbym ,

Would  you  please  give  me  a  link  of  that  answer  if  convenient  ?

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#24 2016-06-13 20:26:09

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

Re: Probability --- triangles

Hi  bobbym ,

The  answer  of  13/120  was  given  by  Andrew S  at  the  4th  answer .
But  I  can't  trace  how  he  got  this  value .
In  fact  I  prefer  to  accept  thickhead 's  result , i.e.  1/10 .

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#25 2016-06-13 20:31:31

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

Re: Probability --- triangles

Update: I have some confirmation on the 1 / 10 answer now in 2 ways.


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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