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#1 2017-03-05 15:36:14

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

Geometric probability----circles

Inside  a  circle  E  there  are  2  smaller  circles  A  and  B  ,both with  radius  being 1/2  of  that  of  E.  Both  A  and  B  can  move  freely  inside  E .If  a point  is  chosen  randomly  on  E , find  the  probability  that  the  point  lies  inside  A  and  B  at  the  same  time.
Will  the  answer  be  simply  1/4 * 1/4 =  1/16  ?

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#2 2017-03-06 04:30:03

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

Re: Geometric probability----circles

Hi;

I am getting results that suggest the answer is closer to 1 / 9. To check, are both of the smaller circles fully inside the larger circle?

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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#3 2017-03-06 14:52:23

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

Re: Geometric probability----circles

Hi  bobbym ,

The  smaller  circles  must  be  fully  inside  the  big  one , they 
cannot  pass  through  its  circumference .
If  the  answer  is  closer  to  1 / 9 , then  the  result  will  be 
the  same  as  the  case  for  rectangles .

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#4 2017-03-06 15:25:31

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

Re: Geometric probability----circles

That I am not sure about. Where as the intersection of two rectangles was another rectangle, the area of the intersection of two circles involves trig functions. So, while I think the answer is close to 1 / 9 or 1 / 10, I do not think it equals a fraction.

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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#5 2017-03-06 18:50:43

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

Re: Geometric probability----circles

{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 2017-03-07 15:34:03

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

Re: Geometric probability----circles

Thanks  thickhead ,

How  did  you  get  the  result  1/4 - 4 / 3 π^2 , from  integrals  ?

I  know  that  it  is  much  more  complicated  to  find  the  area  of  intersection  of  2  circles , but  it  will  be    necessary   to  solve  this  problem .

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#7 2017-03-07 17:54:32

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

Re: Geometric probability----circles

{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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#8 2017-03-08 15:48:45

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

Re: Geometric probability----circles

Hi  thickhead ,

Then  what  will  be  P  for  3  moving  circles  ?

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#9 2017-03-08 16:10:56

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

Re: Geometric probability----circles

For 3 moving circles  it is 0.0663554.
For 4 moving circles  it is 0.0433163
For 5 moving circles  it is 0.0305428.
For 6 moving circles  it is 0.0227118.
For 10 moving circles  it is 0.00948101

Last edited by thickhead (2017-03-08 16:11:20)

{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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#10 2017-03-09 17:15:31

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

Re: Geometric probability----circles

Thus  the  probability  for  various  polygons  will  be  :

                                 no.  of  moving  polygons                             
   P        ||      1            ||                   2                       ||          3              ||
________________________________________________________
circles   ||  1/4 = 0.25  || 1/4 - 4 / 3 π^2 = 0.1149    ||     0.0663            ||

squares  ||  1/4 = 0.25 ||         1/9  = 0.111                ||   1/16 = 0.0625  ||

triangles ||  1/4 = 0.25 ||         1/10 = 0.1                   ||   1/21 = 0.0476  ||

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#11 2017-03-11 16:23:29

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

Re: Geometric probability----circles

Related  problem  ( I )

Inside  a  circle  E  with  radius   e   unit  there  are  2  smaller  circles  A  and  B   with  radius   a  unit  and  b  unit   respectively  where   a ≤ e   and   b ≤ e . Both  A  and  B  can  move  freely  inside  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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#12 2017-03-16 15:15:35

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

Re: Geometric probability----circles

Related  problem  ( II )

On  the  surface  of   a  sphere  E   with  circumference  being   1   unit   there  are  2  circles 
A  and  B  both  with   circumferences  being  1/2  unit .  Both  A  and  B  can  move  freely  and  randomly  on  the  surface  of  E .
If  a  point  is  chosen  randomly  on  the  surface  of  E , find  the  probability  that  the  point  lies  inside  A  and  B  ( referring  from  their  minor  portions )  at  the  same  time .

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#13 2017-03-19 15:34:20

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

Re: Geometric probability----circles

Related  problem  (III)

2  circles  A  and  B  both  with  radius  1 unit  rotate  freely  and  randomly 
outside  a  circle  E  also  with  radius  1  unit  on  its  circumference .
Find  the  expected  value  of  the  overlapping  area  of  A  and  B .

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#14 2017-03-27 15:22:24

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

Re: Geometric probability----circles

For  the  original  problem , what  will  be  the  probability  if  both  A  and  B  cannot  get 
through  the  centre  of  E  ?

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#15 2017-03-27 21:09:55

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

Re: Geometric probability----circles

Last edited by thickhead (2017-04-02 20:45:22)

{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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#16 2017-03-28 16:37:18

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

Re: Geometric probability----circles

Thanks  thickhead ,

What  will  be  the  probability  if  A  cannot  get  through  the  centre  of  E  but  B  can  ?

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#17 2017-03-28 20:41:50

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

Re: Geometric probability----circles

Related  problem  (III)

2  circles  A  and  B  both  with  radius  1 unit  rotate  freely  and  randomly
outside  a  circle  E  also  with  radius  1  unit  on  its  circumference .
Find  the  expected  value  of  the  overlapping  area  of  A  and  B .

Can you clarify the question. Are the centers of A and B on the circumference of E or does each of them just intersect with E at one point ( circumference to circumference)?

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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#18 2017-03-29 15:31:04

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

Re: Geometric probability----circles

Hi  bobbym ,

I  mean  the  circumferences  of  A  and  B  rotate  on  the  circumference  of  E . ( Just  touch 
at  1  point . )

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#19 2017-03-29 17:45:33

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

Re: Geometric probability----circles

Was an answer given for this question yet?

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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#20 2017-03-30 15:24:26

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

Re: Geometric probability----circles

Hi  bobbym ,

No , I  don't  know  the  answer .

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#21 2017-03-30 15:27:23

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

Re: Geometric probability----circles

Thanks. I have an answer but until I get the same answer using another way...

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 2017-04-02 15:45:27

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

Re: Geometric probability----circles

mr.wong wrote:

Related  problem  (III)

2  circles  A  and  B  both  with  radius  1 unit  rotate  freely  and  randomly 
outside  a  circle  E  also  with  radius  1  unit  on  its  circumference .
Find  the  expected  value  of  the  overlapping  area  of  A  and  B .

Although  problem (III)  states  that  both  A  and  B  are  movable .
But  if  one  of  them  , say  A  is  fixed  touching  E  at  a  certain 
point , with  only  B  movable ,  the  answer  should  be  the  same .
Thus  the  problem  will  become  simpler .
A  possible  way  to  solve  the  problem  is  to  consider  the  angle 
between  the  centers  of  the  3  circles .

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#23 2017-04-02 20:14:33

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

Re: Geometric probability----circles

the  answer  should  be  the  same.

I agree and have been using that idea the whole time. Unfortunately, the computation of the integrals or sums is very difficult. The closest I can get right now is to say the expectation is about .43

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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#24 2017-04-02 21:00:15

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

Re: Geometric probability----circles

mr.wong wrote:

Thanks  thickhead ,

What  will  be  the  probability  if  A  cannot  get  through  the  centre  of  E  but  B  can  ?

Product of the 2 probabilities integrated over the region gives the required probability.0.08618
http://www.wolframalpha.com/widgets/view.jsp?id=8ab70731b1553f17c11a3bbc87e0b605

{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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#25 2017-04-03 16:37:39

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

Re: Geometric probability----circles

Thanks  bobbym  and  thickhead ,

For  Related  problem  (III) , should  there  be  a  formula  to  calculate 
the  area  of  overlapping  area  of  A  and  B   if  the  angle  between 
the  3  centers  being  x  degrees  where  0 ≤ x < 60  ?

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