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Pair Ponder (Posted on 2014-08-07) Difficulty: 3 of 5
Three points are chosen randomly and independently on a circle.

Determine the probability that each of the three pairwise distances between the points is less than the radius of the circle.

No Solution Yet Submitted by K Sengupta    
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Some Thoughts simulation verification Comment 2 of 2 |
The position on the circle is given in units of one circumference and is measured along the circumference. Therefore if the distance between two points is more than one half, its complement is taken. The distance along the arc must be under 1/6 circumference for the chord distance to be less than one radius.

DefDbl A-Z
Dim crlf$, ptpos(3)
Function mform$(x, t$)
  a$ = Format$(x, t$)
  If Len(a$) < Len(t$) Then a$ = Space$(Len(t$) - Len(a$)) & a$
  mform$ = a$
End Function

Private Sub Form_Load()
 ChDir "C:\Program Files (x86)\DevStudio\VB\projects\flooble"
 Text1.Text = ""
 crlf$ = Chr(13) + Chr(10)
 Form1.Visible = True
 DoEvents
 
 For trial = 1 To 12000
   For pt = 1 To 3
     ptpos(pt) = Rnd(1)
   Next pt
   good = 1
   For pt1 = 1 To 2
     For pt2 = pt1 + 1 To 3
       dist = Abs(ptpos(pt1) - ptpos(pt2))
       If dist > 0.5 Then dist = 1 - dist
       If dist > 1 / 6 Then good = 0: Exit For
     Next
   Next
   If good Then goodct = goodct + 1
   If trial Mod 120 = 0 Then
     Text1.Text = Text1.Text & mform(goodct, "#######") & mform(trial, "########") & mform(goodct / trial, "##0.000000") & mform(trial / goodct, "###.0000") & crlf
     DoEvents
     Randomize Timer
   End If
 Next
 
 
 Text1.Text = Text1.Text & crlf & "done"
End Sub

It finds, in one run:

 939   12000  0.078250 12.7796
 
That is, the condition held in 939 out of 12,000 trials or 0.07825 of the time, or 1/12.7796.

Another run of the program:

1018   12000  0.084833 11.7878

and another:

998   12000  0.083167 12.0240

  Posted by Charlie on 2014-08-07 13:29:43
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