geometric probability ---- squares

bobbym · 2016-06-16 17:21:34

I did 3 moving squares and got 1 / 16 as predicted.

mr.wong · 2016-06-17 15:42:13

Hi bobbym ,

I wonder why thickhead can get the formula for n squares directly
with double integration while proved by mathematical induction is not
necessary .

bobbym · 2016-06-17 15:44:13

I did not see where that was done, can you point the post out?

mr.wong · 2016-06-17 21:25:52

Hi bobbym ,

It was in # 74 .

bobbym · 2016-06-18 02:37:45

Hi;

Okay thanks. I did not see that one.

mr.wong · 2016-06-21 16:26:40

thickhead wrote:

mr.wong wrote:
Thanks bobbym , you are right .
By formula , we have the probability of the point lies
within the axis of 1 side ( say horizontal side ) of the
common portion of A and B to be 1/3 . Similarly ,
for the vertical side , P also = 1/3 . Thus combinely
P of the point lies within both A and B will be 1/9 .
Will the answer be the same for other polygons , say
rhombus under similar conditions ?
Could you elaborate on the formula?

Hi thickhead ,

Previously you have asked about the formula I used to find the
probability involving squares . You can find it in the thread
"geometric probability ---segments " # 22 .

thickhead · 2016-06-29 18:11:52

mr.wong wrote:

Related problem ( I ) : To find the volume of a polyhedron
Let X denotes a polyhedron with 6 vertices PQRSTU ,
11 sides and 7 faces :
(1) base PQRS being a square with sides 1/2 unit .
(2) Δ PTS with PT = 1/12 unit ( in fact sq.unit)
where TP is perpendicular to the base .
(3) Δ PTQ being congruent to Δ PTS .
(4) Δ RUQ with RU = 1/8 unit ( or sq.unit )
where RU is perpendicvlar to the base .
(5) Δ RUS being congruent to Δ RUQ .
(6) Δ TQU .
(7) Δ TSU being congruent to Δ TQU .
How to find the volume of X ?

Hi mr.wong,
Did you get this volume as 5/288 ?
http://onlinemschool.com/math/assistance/vector/pyramid_volume/
use this to calculate pyramid volume in proper fractions .
you can use the following for calculating polyhedron volume directly but the answer will be in decimals.
http://www.staff.amu.edu.pl/~pawel/volume/calculator.html

Last edited by thickhead (2016-06-29 18:12:23)

mr.wong · 2016-06-30 16:06:25

Thanks thickhead ,

The answer 5/288 seems to be incorrect .
In fact the aim of finding the volumes of those
polyhedrons is to provide a method to solve the
problem involving 2 moving triangles but I failed .
Thus all the effort were wasted and I don't think
I will spend time dealing with the polyhedron any
more .

mr.wong · 2016-07-05 15:30:06

Related problem :

Inside a square E with co-ordinates ( 0,0) , (2,0), (2,2) and
(0,2) there are 2 smaller squares A and B both with length of sides being 1 unit and parallel with E . A can move freely and uniformly inside E but keep parallel with E during moving ; while B stays fixed in E with co-ordinate of its south-west vertex being ( x ,y ) . A point is chosen randomly on E .
(1) Find the probability that the point lies inside A and B
at the same time .
(2) If the probability = 1/9 , solve x and y .

thickhead · 2016-07-05 21:03:43

Last edited by thickhead (2016-07-08 03:01:55)

mr.wong · 2016-07-06 16:45:32

Hi thickhead ,

Why P is not symmetric in x and y ?
For x = y can the results be expressed in fractions ?
Should there be totally 4 answers ( or more ? ) for (x , y ) ?
We know that for (x , y ) = ( 0,0) , ( 1,0) , (0,1) or (1,1)
P will have smallest value while for (x,y) = (1/2,1/2)
P will have greatest value . ( This is the case when B is
located at the centre of E .) 1/9 should be the average value
of P .

thickhead · 2016-07-06 18:24:24

P is symmetric in x and y. You replace x by y and y by x , the formula remains the same.You get infinite points (x,y) for a given value of P say 1/9. The average works out to be 4/27.that is because we integrate the average probability when south west point is at (x,y). This means the average of average which is not correct.

Last edited by thickhead (2016-07-07 20:47:28)

mr.wong · 2016-07-07 15:52:26

Hi thickhead ,

Your formula in #85 seems not symmetric in x and y .
If P = 1/9 , I think there should be 4 answers for ( x, y) .
In fact the aim of this related problem is to find whether in the
problem involving 2 moving squares , 1 of them can be replaced
by a fixed square with same length and yields the same answer .
( of course the location of the fixed square cannot be random .)

thickhead · 2016-07-08 05:30:26

The maximum value of the probability 9/64 occurs at(1/2,1/2) For P=1/9

x	                y1   	       y2
0.896746024	     0.5	       0.5
0.788675135	     0.788675135  	0.211324865
0.7	                0.852167445	0.147832555
0.6	                0.886522185	0.113477815
0.5	                0.896746024	0.103253976
0.4	                0.886522185	0.113477815
0.3	                0.852167445	0.147832555
0.2	                0.776765834	0.223234166
0.211324865	       0.788675135  	0.211324865
0.11	                0.581593877	0.418406123
0.103253976	          0.5          0.5

thickhead · 2016-07-08 22:31:27

Math Is Fun Forum

#76 2016-06-16 17:21:34

Re: geometric probability ---- squares

#77 2016-06-17 15:42:13

Re: geometric probability ---- squares

#78 2016-06-17 15:44:13

Re: geometric probability ---- squares

#79 2016-06-17 21:25:52

Re: geometric probability ---- squares

#80 2016-06-18 02:37:45

Re: geometric probability ---- squares

#81 2016-06-21 16:26:40

Re: geometric probability ---- squares

#82 2016-06-29 18:11:52

Re: geometric probability ---- squares

#83 2016-06-30 16:06:25

Re: geometric probability ---- squares

#84 2016-07-05 15:30:06

Re: geometric probability ---- squares

#85 2016-07-05 21:03:43

Re: geometric probability ---- squares

#86 2016-07-06 16:45:32

Re: geometric probability ---- squares

#87 2016-07-06 18:24:24

Re: geometric probability ---- squares

#88 2016-07-07 15:52:26

Re: geometric probability ---- squares

#89 2016-07-08 05:30:26

Re: geometric probability ---- squares

#90 2016-07-08 22:31:27

Re: geometric probability ---- squares

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