geometric probability--segments

mr.wong · 2016-01-18 15:28:36

Hi bobbym ,

For example , let a = b = 1/10 , then your formula will get

P = ( 1/100 - 2/10 + 3/100 - 3/10 + 1 ) / ( - 27/10 )

= [( 1 +3 +100 - 20 - 30 ) / 100 ] / ( -27/10 )

= ( 54/100 ) *( - 10 / 27 )

= - 1/ 5

Another example , let a = b = 2/5 ,

my formula yields P = 28/135
while your formula yields P = 1/5 = 27/135 ;
which one is more correct ?

bobbym · 2016-01-18 17:32:19

Hi;

The formula I posted was for

when a>b:

when a = b

mr.wong · 2016-01-20 14:14:51

Hi bobbym ,

Your second formula is too complicated , I can't even
check its validity for a = b = 2/5 .
Don't spend time on your formula anymore ! Just use
my formula for further investigation if you have checked its validity .

Relentless · 2016-01-20 14:52:04

Hi;

I do not know what this thread is about, but the bottom equation can be simplified if you define x=a=b:

Which is 28/135 when x=a=b is 2/5

bobbym · 2016-01-20 17:11:59

Hi;

Don't spend time on your formula anymore ! Just use
my formula for further investigation if you have checked its validity .

If you do not want it then there is no point but the reason I favor replacing non analytical functions like min and max with algebraic expressions is because they are much easier to derive more information from. For instance, relentless' simplification. Also we can integrate and differentiate algebraic expressions to derive means and standard deviation and other moments, this will be difficult with min and max.

mr.wong · 2016-01-22 15:48:01

Hi Relentless ,

Thanks for your reply ! Though your formula is valid
for x=a=b=2/5 , but if you substitute x=a=b by 3/5 ,
you will get a not so correct result 9/20 instead of
7/15 .

mr.wong · 2016-01-22 15:51:51

Hi bobbym ,

The use of function with min and max will save much labours , for it applies to every possible conditions but not just certain conditions .
I pay respect to your favour and hard works , but time and spirit should be spent on the right place .
Later I shall discuss some related topics in a new thread .

Relentless · 2016-01-22 16:08:52

Hi mr wong, bobbym's formula also gives 9/20 for a=b=3/5. I have confirmed graphically that my formula and his are equivalent - if the answer should be 7/15 then both formulas are definitely wrong. In fact, no positive number will give an answer of 7/15 ... I am getting a=b~-0.869 as the only solution for 7/15 (the exact form is complicated).

But again, I am afraid I do not know what we are talking about! Haha

Last edited by Relentless (2016-01-22 16:11:59)

bobbym · 2016-01-22 17:27:07

I agree that 7 / 15 is correct. The formulas I posted are only valid for 0 < a < 1/2, a >= b. Putting 3 / 5 in them violates that condition.

thickhead · 2016-06-22 05:34:07

I got 28/135 for a=b=3/5 as well as 7/15 for 2/5. I shall bring it out later.

thickhead · 2016-06-22 20:22:26

mr.wong · 2016-06-22 23:02:26

Hi thickhead ,

Should it be 28/135 for a=b=2/5 as well as
7/15 for 3/5 ? ( # 35 )

thickhead · 2016-06-22 23:39:55

Last edited by thickhead (2016-06-23 04:37:41)

mr.wong · 2016-06-23 16:54:09

Hi thickhead ,

It seems you have developed another valid formula
for 2 moving segments which looks quite different
with mine , though it appears a bit too complicated .
Moreover , your formula seems not symmetric for
a and b . More data should be necessary for proving
its validity . ( I suggest you take reference to my formula .)
However , after your formula has been confirmed
I hope you can modify it to one applicable for 3 segments ( variables ) .

thickhead · 2016-06-23 17:45:43

Last edited by thickhead (2016-06-23 21:29:04)

thickhead · 2016-06-25 00:15:57

bobbym wrote:

Hi;
The first bit of real results in a long time of hard work.
If we denote the length of the three segments that will slide along E, S1,S2 and S3.
1) Then:
That covers some of the cases, I have more but need time to check them...

This equation is surprisingly synonymous with what I derived. bobbym is very mysterious. Never discloses how he derived.

bobbym · 2016-06-25 03:37:06

No mystery, just a set of rules.

1) When posting to a student on a homework question, I will sometimes give a method but not the answer and more times the answer without a method. We believe that a problem solver must be able to back engineer a method from the solution. This is a very valuable skill. The student has to do at least part of the work if he is to learn anything.

2) Sometimes the OP just is looking for an answer. With that answer, mr.wong just wanted to verify his own work so a solution was not necessary.

3) It is easy in my opinion for someone doing EM to get the answer. Whereas in classical mathematics they thrash around with algebraic symbols until an answer finally shows itself, in EM we often have the answer but not the method. That is why rule 1 is so important.

4) This is not a rule but a fact of life, sometimes I can easily get the answer but am not smart enough to develop a method that will satisfy a math type.

thickhead · 2016-06-25 04:30:23

Bobbym
Thank you for clarification.

thickhead · 2016-06-25 21:57:39

Last edited by thickhead (2016-06-27 01:45:02)

thickhead · 2016-06-26 04:12:21

thickhead · 2016-06-26 04:36:30

mr.wong · 2016-06-26 21:46:11

Hi thickhead ,

It is quite astonished that for problem involving
2 segments , both formula which looks quite different
are in fact identical . ( In # 44, the term (1-c) should be
removed from my formula ! )

For my own convenience I have re-written your formula
for 2 variables in the following .( Where
m denotes min [min (a,1-a),min (b,1-b) ] and
n denotes max [min (a,1-a),min (b,1-b)] , thus there
is no need to consider the order of m and n . )

P(2) = [mn(1-n)- m^3 /3 ] / (1-a)(1-b)
= min ( a, 1-a, b, 1-b )* max [min (a,1-a),min (b,1-b)]
* { 1- max [min (a,1-a),min (b,1-b)]}
- min ( a, 1-a, b, 1-b ) ^3 / 3 / (1-a) (1-b)

thickhead · 2016-06-26 21:53:00

Hi mr.wong,
I would like the basis on which your formula was derived. Is it in existence for a long time ? I have started the subject from scratch and had no exposure to it earlier.

mr.wong · 2016-06-27 23:06:13

Hi thickhead ,

In fact I developed the formula more than 20 years
ago ! I haven't publish it to the public until last year
in a Chinese math . forum . I got the formula through
solid geometry and had not expected that the related
problem can also be solved with integration and derive
a formula which looks quite different with mine ! I hope
my formula can be modified and generalized to one involving 3 variables .

thickhead · 2016-06-27 23:53:03

Hi mr.wong,
You said my formula can be modfied to yours but how do you explain max(0,1-a-b) term ?
1-a-b is the gap left over if A and B are joined end to end. and min(1-a,1-b) is the gap left if A and B are superposed.
but I am unable to interpretation of those terms in the formula. One more thing where these ideas can be implemented in practice?

Math Is Fun Forum

#26 2016-01-18 15:28:36

Re: geometric probability--segments

#27 2016-01-18 17:32:19

Re: geometric probability--segments

#28 2016-01-20 14:14:51

Re: geometric probability--segments

#29 2016-01-20 14:52:04

Re: geometric probability--segments

#30 2016-01-20 17:11:59

Re: geometric probability--segments

#31 2016-01-22 15:48:01

Re: geometric probability--segments

#32 2016-01-22 15:51:51

Re: geometric probability--segments

#33 2016-01-22 16:08:52

Re: geometric probability--segments

#34 2016-01-22 17:27:07

Re: geometric probability--segments

#35 2016-06-22 05:34:07

Re: geometric probability--segments

#36 2016-06-22 20:22:26

Re: geometric probability--segments

#37 2016-06-22 23:02:26

Re: geometric probability--segments

#38 2016-06-22 23:39:55

Re: geometric probability--segments

#39 2016-06-23 16:54:09

Re: geometric probability--segments

#40 2016-06-23 17:45:43

Re: geometric probability--segments

#41 2016-06-25 00:15:57

Re: geometric probability--segments

#42 2016-06-25 03:37:06

Re: geometric probability--segments

#43 2016-06-25 04:30:23

Re: geometric probability--segments

#44 2016-06-25 21:57:39

Re: geometric probability--segments

#45 2016-06-26 04:12:21

Re: geometric probability--segments

#46 2016-06-26 04:36:30

Re: geometric probability--segments

#47 2016-06-26 21:46:11

Re: geometric probability--segments

#48 2016-06-26 21:53:00

Re: geometric probability--segments

#49 2016-06-27 23:06:13

Re: geometric probability--segments

#50 2016-06-27 23:53:03

Re: geometric probability--segments

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