Math Is Fun Forum

mr.wong

Member

Registered: 2015-12-01

Posts: 252

probability --- arcs

Inside the circumference E with length 1 unit of a circle ,
there are 2 arcs A with length 1/3 unit and B with length
1/2 unit . Both A and B can move freely along E , but there
is a point X at E such that A can get through X while B
cannot . Find the expected length of the overlapping portion
of the 2 arcs .

bobbym

bumpkin

From: Bumpkinland

Registered: 2009-04-12

Posts: 109,606

Re: probability --- arcs

Hi;

but there is a point X at E such that A can get through X while B cannot

I do not understand what this means, can you explain a bit more?

---

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 --- arcs

Hi  bobbym ,

It  means  that  there  is  a  fixed  point  X  lying  in  E  which  blocks  B  from 
getting  through  it , thus  B  has  to  return  its  direction  of  moving ,  while  A 
is  not  affected  by  the  existence  of  X , A  can  just  get  through  it  in  moving .

bobbym

bumpkin

From: Bumpkinland

Registered: 2009-04-12

Posts: 109,606

Re: probability --- arcs

Hi;

So, it sort of bounces off that point?

---

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 --- arcs

Hi  bobbym ,

When  one  end  of  B  reaches  the  point  X  , then  it 
will  stop  there .  ( also  for  X ) If  B  moves  again  then  it  can  only  move  backwards . Whenever  one   end  reaches  X  it  will  stop  again . Thus  X  will  never  get  inside  of  B .

mr.wong

Member

Registered: 2015-12-01

Posts: 252

Re: probability --- arcs

The  answer  is  just  simply  be  1/3 * 1/2 = 1/6  un ,
the  point  X  plays  no  role  in  this  problem  and  can 
be  neglected .
The  case  for  an  arc  which  can  be  blocked  by  a 
point  X  in  the  circumference  is  just  equivalent  to  the  case  for  a  small  segment  moving  freely  inside  a  large  segment  but  must  keep  entirely  inside  it .
While  an  arc  not  affected  by  the  existence  of  X  is 
equivalent  to  a  small  segment  moving  in  a  large 
segment  and  permitted  to  exceed   both  ends  of  it  .
(  of  course  still  keep  touching  with  it  ) 
For  cases  involving  only  one  arc  which  can  be  blocked  by  X  ,
the  existence  of  X  can  be  neglected  ,
but  it  will  be  different  if  there  are  more  than  1  such
arcs .  For  example  , if  there  is  an  additional  arc  C 
,say  also  with  length  1/2  unit  and  moving  inside  E , which  can  also  can  be  blocked  by  X . Then  the  expected  length  of  the  overlapping  portion  of  A , B 
and  C  will  be  1/3 * 1/3 = 1/9  un .  ( by  formula )
In  this  case  the  existence  of  X  cannot  be  neglected .
Otherwise  if  arc  C  can  get  through  X  during  moving
, then  the  answer  will  be  1/2 * 1/2 * 1/3 = 1/12  un ,
in  this  case  the  point  X  can  also  be  neglected .

thickhead

Member

Registered: 2016-04-16

Posts: 1,086

Re: probability --- arcs

Does E form the complete circumference of the circle or only part of it? I thought E is also an arc.

---

{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 --- arcs

Hi  thickhead ,

E forms the complete circumference of the circle .

thickhead

Member

Registered: 2016-04-16

Posts: 1,086

Re: probability --- arcs

Hi mr.wong,
Now I understand the situation. when there is no endlimit for the arc every point on the circumference has the same exposure to the arc and uniform probability at every point. On any addition of similar arc the probability of remaining on both is multiplied by each probability. However when there is an arc with end limit. Some points have more exposure and some less exposure but average probability is the same as limitless arc. when arc  with stop at X is added interference of B and c produces probability of

this 1/3 has nothing to do with the first 1/3 of free arc A, a confusion created by your cunning narration.In fact if anther arc of length 1/2 stopping at X is added the probability would be 1/3*1/12=1/36

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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.}