[ExI] isaac's number

[BillK]

On 1 August 2017 at 08:17, BillK wrote:
> On 1 August 2017 at 07:28, spike  wrote:
>>
>> Generate three random points on a square plane to create a triangle.  What
>> is the probability that a fourth randomly-generated point will be inside the
>> triangle?
>>
>> He and I wrote sims and both get the same answer to four significant digits:
>> 0.0764 but neither of us know how to find it in closed form.  My intuition
>> suggested that number should have been 0.125 but it differs from our
>> sim-generated Isaac’s number by about phi.  Why phi?  That hasta be a
>> coincidence, ja?  Come on math geeks, let us reason together, shall we?
>>
>
>
> I think this gives the solution.
> It just needs to be translated into English!   :)
>
> <http://mathworld.wolfram.com/SquareTrianglePicking.html>
>
>


In English,  :)
you need to calculate the average area of the distribution of all
random triangles in a square, between the maximum of half and zero.
(That's what Wolfram is doing). The result comes out at 11/144. So
that is the chance of a fourth random point falling within a random
triangle.

BillK