Probability --- triangles

mr.wong · 2016-05-30 15:34:48

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 ?

mr.wong · 2016-06-06 14:58:37

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 .

bobbym · 2016-06-06 17:35:01

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?

mr.wong · 2016-06-06 21:10:26

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 .

thickhead · 2016-06-07 00:39:02

thickhead · 2016-06-07 17:17:28

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

mr.wong · 2016-06-07 22:24:53

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 .

thickhead · 2016-06-09 21:24:21

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

mr.wong · 2016-06-09 21:37:04

Hi thickhead ,

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

thickhead · 2016-06-09 22:20:20

thickhead · 2016-06-09 22:45:41

mr.wong · 2016-06-09 23:51:08

Thanks thickhead again ,

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

thickhead · 2016-06-10 18:55:19

thickhead · 2016-06-10 19:05:02

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)

mr.wong · 2016-06-11 15:40:32

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 !

bobbym · 2016-06-11 17:00:28

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.

mr.wong · 2016-06-11 20:04:09

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 .

bobbym · 2016-06-11 20:54:36

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.

thickhead · 2016-06-12 05:06:40

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)

mr.wong · 2016-06-13 14:49:07

Hi thickhead ,

Then what should be the results ?

bobbym · 2016-06-13 15:01:41

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.

thickhead · 2016-06-13 16:18:37

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.

mr.wong · 2016-06-13 18:56:21

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 ?

mr.wong · 2016-06-13 20:26:09

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 .

bobbym · 2016-06-13 20:31:31

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

Math Is Fun Forum

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

Probability --- triangles

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

Re: Probability --- triangles

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

Re: Probability --- triangles

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

Re: Probability --- triangles

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

Re: Probability --- triangles

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

Re: Probability --- triangles

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

Re: Probability --- triangles

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

Re: Probability --- triangles

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

Re: Probability --- triangles

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

Re: Probability --- triangles

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

Re: Probability --- triangles

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

Re: Probability --- triangles

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

Re: Probability --- triangles

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

Re: Probability --- triangles

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

Re: Probability --- triangles

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

Re: Probability --- triangles

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

Re: Probability --- triangles

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

Re: Probability --- triangles

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

Re: Probability --- triangles

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

Re: Probability --- triangles

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

Re: Probability --- triangles

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

Re: Probability --- triangles

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

Re: Probability --- triangles

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

Re: Probability --- triangles

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

Re: Probability --- triangles

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