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#1 2016-09-14 15:02:42

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

Probability of meeting

A  boy  and  a girl  have  dated  to  meet  at  a  place .
They  will  arrive  there    randomly  within  60  minutes   
and   are  willing  to  wait  for  one  another   for  20 mins.
What  is  the  probability  of  their  meeting  at  the  place?

( Solution  illustrated  with  a  diagram  will  be  appreciated .)

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#2 2016-09-14 17:08:44

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

Re: Probability of meeting

Hi;


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 2016-09-14 23:01:19

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

Re: Probability of meeting

Thanks  bobbym ,

Your  answer  is  correct !
Now  let  us  ask  another  question :  What  will  be  the  expectation  of  the 
waiting  time  of  any  one  of  them , say  the  boy ? 

Your  diagram  may  be  divided  into  3  portions  . If  the  blue  portion 
is  bisected  by  the  diagonal  ( the  curve  y = x ) then  it  becomes  4 
portions .
Let  A , B , C  and  D  denote  the  4  portions  one  by  one  from  the 
south - east  triangle  to  the   north - west  one . 
Let  the  boy  arrives  at  x  minutes  and  the  girl  arrives  at  y  minutes .
For  A , x - y ≥  20  .  The  boy  does  not  see  the  girl  , but  still  expecting  she  will  come  latter .   He  will  wait  for  min ( 20 , 60-x )  minutes .
For  B , x - y  ≤  20  , the  boy  need  not  wait  , i.e.  he  will  wait  for  0  minute .
For  C , y - x  ≤  20 ,  the  boy  will  wait   for  y-x  minutes .
For  D , y - x  ≥  20 ,  the  boy  will  keep  waiting  for  20  minutes .   

How  should  we  continue  ?

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#4 2016-09-15 06:20:31

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

Re: Probability of meeting

Hi;

A  boy  and  a girl  have  dated  to  meet  at  a  place .
They  will  arrive  there    randomly  within  60  minutes   
and   are  willing  to  wait  for  one  another   for  20 mins.
What  is  the  probability  of  their  meeting  at  the  place?

Supposing the boy has to wait 28 minutes? What do you want to do with that possibility? Would the boy have left after 20 minutes? That would make the maximum waiting time for him of 20 minutes.


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 2016-09-15 14:56:08

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

Re: Probability of meeting

Hi  bobbym ,

You  are  right . The  maximum  waiting  time  for  the  boy  is  20  mins .
He  will  left  immediately  after  20  mins . even  he  knows  that  the  girl  will 
come  8  mins.  later .

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#6 2016-09-15 15:15:19

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

Re: Probability of meeting

Also, when the girl gets there and waits 20 minutes she will leave, now the boy comes, he does not know she was there so he waits 20 minutes also and then leaves. Is that how you want to handle that situation?


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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#7 2016-09-16 15:13:00

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

Re: Probability of meeting

Hi  bobbym ,

Yes , that's  the  case .

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#8 2016-09-16 15:17:41

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

Re: Probability of meeting

To  find  the  expectation  of  the  boy's   waiting  time  ,  we  may  use  a 
3-dimensional  diagram  with  base  of  bobbym's  diagram  and  the 
height   at   z-axis  showing  the  corresponding  waiting  time .

(1)  For  portion ( A ) ,  it  may  be  re-divided  into  3  regions .
     (i)  region  A1   denotes  the  triangle  (20,0) , (40,0) and  (40,20)
     with  height   being  20 ( minutes)  and  area  =  20 * 20 *  1/2  = 200  .
     Thus  its  volume  =  20 *  200  = 4000 .

  (ii)  region  A2  denotes  the  square  (40,0) , (60,0) , (60,20)  and   (40,20)  with  heights  being  20  at  one  side  and  0  at  the  opposite 
side  .  Thus  the  volume  of   this   prism   =  20* 20 * 1/2 * 20  =   4000 .

(iii)  region  A3  denotes  the  triangle  (40,20), (60,20) and (60,40) .
     with  the  corresponding  pyramid  with  height  being  20 .
    Thus  the  volume  =  1/3 * 20 * 20  * 20 * 1/2  =  4000 / 3  .
Thus  total  volume  over  A  will  be  4000 + 4000 + (4000 / 3 ) 
=  8000 + (4000 / 3 ) 


(2)  For  portion  (B) , since  the  corresponding  height  =  0  , 
  therefore  the  corresponding  volume  also =  0  .

(3)  For  portion (C) , it  may  also  be  divided  into  3  regions  .
  (i)  region  C1  denotes  the  rectangle  with  length  ( 0,20) and (40,60) ,
    =  √2   * 40  .     and  width  =    √2   * 10 .
Volume   of   corresponding  prism  =  20 * √2   * 10 * 1/2  * √2   * 40
=  8000

(ii) region  C2  being  a  triangle  at  one  side  of  the  rectangle  with  area 
√2   * 10   *  √2   * 10  * 1/2  =  100  . Thus  volume   of  the  corresponding   pyramid  =  1/3 * 20 * 100  = 2000 / 3

(iii)  region C3  being  another  symmetric  triangle  at  the  other  side  of 
the  rectangle . Its  area  and  volume  of  the  corresponding  pyramid  are 
identical  to  those  of  region  C2 . Thus  the  corresponding  volume  is 
also  2000 / 3  .
Thus  total  volume  over  C  will  be  8000  + ( 2000 / 3)  * 2
= 8000 + (4000 / 3 )     (  same  as  A .)

(4)   For  portion  D , its  area  =  40 * 40 * 1/2  = 800 .  Height  of  the 
corresponding  prism  =  20  ,  thus  its  volume  =  800 * 20  =  16000 .

Thus  the  total  volume  over  the  base  =  16000 +  8000 / 3  + 16000
= 32000 +  8000 / 3  =  104000 / 3

So  the  expectation  of  the  waiting  time  of  the  boy 
=  104000  / ( 3  * 60  *  60 ) 
=  260 /  27  mins . 
=   9 +  17/ 27  mins .    (  about  9.63  mins . )

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#9 2016-09-16 15:23:09

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

Re: Probability of meeting

Hi;

I am not getting that.


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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#10 2016-09-16 21:39:41

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

Re: Probability of meeting

Hi  bobbym ,

What  result  did  you  get  ? 
My  result  may  be  incorrect  for  I  have  not  checked  it  carefully , 
but  the  procedure  should  be  ok .

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#11 2016-09-16 23:16:15

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

Re: Probability of meeting

Hi  bobbym ,

I  have  found  where  I  made  a  mistake !
The  total  volume  over  the  base  should  be 
16000 +  8000 + ( 4000 / 3 )  +  16000 
=  40000 + ( 4000 / 3 ) 
=  124000 / 3

So  the  expectation  of  the  waiting  time  of  the  boy
=  124000  / ( 3  * 60  *  60 ) 
=   310 /  27   mins .
=   11 +  13 / 27   mins.   (  about   11.5  mins )

Hope  this  answer  be  correct  .

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#12 2016-09-17 06:18:48

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

Re: Probability of meeting

That is correct! But I did it in 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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#13 2016-09-17 16:52:49

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

Re: Probability of meeting

Hi  bobbym ,

I  am  confused  now  !  It  seems  my  answer  in  # 8   should  be  correct  ! 
(  I  had  not  made  any  mistake . )  In  fact  I  made  a  mistake  in  # 11 .
Please  check  your  work .

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#14 2016-09-17 16:56:58

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

Re: Probability of meeting

The answer in post #11 agrees with my simulation and my analytical solution. I believe it is the correct 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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#15 2016-09-17 20:59:58

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

Re: Probability of meeting

I get 5 minutes average waiting time.


{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 2016-09-18 15:53:06

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

Re: Probability of meeting

Hi  bobbym ,

I  have  checked  my  solution  in  # 8  for  several  times  but  I  can't 
find  where  I  made  a  mistake .

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#17 2016-09-18 15:54:10

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

Re: Probability of meeting

Hi  thickhead ,

Please  show  your  procedure  .

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#18 2016-09-18 19:06:43

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

Re: Probability of meeting

You do not know what the answer is?


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 2016-09-18 22:05:23

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

Re: Probability of meeting

Hi  bobbym ,

No , I  don't .

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#20 2016-09-19 00:15:02

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

Re: Probability of meeting

Hmmm, that is a problem.


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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#21 2016-09-19 18:22:35

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

Re: Probability of meeting

Simple approach:-
From boy's point of view.His arrival time is t.
(a)t=20 to t=40 minutes. the girl may show up at any time between t-20 to t+20. for half this time his waiting time is 0. For the other half it varies from 0 to 20 averaging 10 minutes. For the 2 periods combined the average is 5 minutes.
(b)At t=0 he has to wait for anything from 0 to 20 m.average waiting time=10 m. From t= 0 to t=20 we can assume linear variation from 10 to 5 minutes.
(c)At t=60 waiting time =0. assume linear variation from 5m at t=20 t0 0 at t=60
the average as whole={(10+5)/2*20+5*20+(5+0)/2*20}/60= 5 minutes.

Last edited by thickhead (2016-09-20 18:25:57)


{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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#22 2016-09-20 17:49:24

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

Re: Probability of meeting

Hi  thickhead ,

I  wonder  whether  the  problem  can  be  solved  in  your  way   .
You  should  take  reference  to  bobbym's  diagram . The  weight 
( area )  of  various  regions  are  not  fixed  thus  you  can't  just  take 
simple  average .
E.g. in (a)  from  t = 20  to  t = 40 , only  1/3  but  not  half  the  time 
= 0 . Another  1/3  the  average  time  = 20 
while  the  remaining  1/3  the  average  time = 10  ( ? ) .
Thus  the  combined  average  in (a) should  sad 0 + 20 + 10  ) / 3 = 10  .

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#23 2016-09-20 17:58:31

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

Re: Probability of meeting

Last edited by thickhead (2016-09-20 17:59:36)


{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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#24 2016-09-20 18:30:03

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

Re: Probability of meeting

mr.wong wrote:

Hi  thickhead ,

I  wonder  whether  the  problem  can  be  solved  in  your  way   .
You  should  take  reference  to  bobbym's  diagram . The  weight 
( area )  of  various  regions  are  not  fixed  thus  you  can't  just  take 
simple  average .
E.g. in (a)  from  t = 20  to  t = 40 , only  1/3  but  not  half  the  time 
= 0 . Another  1/3  the  average  time  = 20 
while  the  remaining  1/3  the  average  time = 10  ( ? ) .
Thus  the  combined  average  in (a) should  sad 0 + 20 + 10  ) / 3 = 10  .

Half the time accounts for this. If the girl comes earlier by 0 to 20 minutes his waiting time is 0.If she comes later between 0 to 20 minutes the average waiting time is 10. the average of these two is 5.


{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 2016-09-20 20:01:17

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

Re: Probability of meeting

What about the times the boy must wait 20 minutes. That adds to the average you get of 5 minutes. Does this not suggest that your answer of 5 or less is not 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.

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