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Same side story (Posted on 2017-03-16) Difficulty: 3 of 5
Suppose n (n>1) points are placed at random on a circumference of a circle.
If P(n) denotes the probability that all n points lie on the same side of some diameter - find P(2), P(3) and P(4).

No Solution Yet Submitted by Ady TZIDON    
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Solution re: Corrected solution | Comment 4 of 5 |
(In reply to Corrected solution by Dan Rosen)

A simulation shows that in the first 100 trials, in 50 cases all four points fell in one semicircle, and in the first 10,000 trials, 5,036 cases had all four points in one semicircle.

This seems to indicate that P(4)=1/2 is the correct answer.

The first 100 trials are shown, with the location in degrees shown for each point, followed by the maximum distance apart for successive points (this may be from the last numerically to the first numerically). If this is larger than 180°, this is a hit, as all the points are within the remaining 180° or less. Hits are marked with an *, so the evaluation of a hit may be verified by inspection.

  1  104.24 192.03 208.63 254.00    210.25 *
  2    5.05 108.70 273.86 278.91    165.16
  3   16.33 149.05 255.25 293.22    132.72
  4  134.47 284.57 310.54 346.30    150.10
  5   20.25 131.05 313.72 341.84    182.67 *
  6   19.26 188.95 213.28 276.16    169.69
  7  107.34 168.73 224.17 233.22    234.12 *
  8   94.97 100.56 296.86 298.73    196.29 *
  9   81.67 212.10 327.95 354.99    130.43
 10   87.82 192.19 250.24 352.80    104.38
 11    5.65  38.29 243.42 359.79    205.13 *
 12   36.02  37.09 207.07 287.60    169.98
 13   16.43 102.41 106.48 137.52    238.91 *
 14  108.35 144.49 341.49 352.74    196.99 *
 15   57.76  58.62 100.18 232.77    184.99 *
 16  117.43 147.63 148.60 256.58    220.85 *
 17   66.96  74.72 210.01 227.94    199.02 *
 18   29.06  94.09 164.87 326.06    161.19
 19  104.28 136.40 282.68 330.98    146.27
 20   35.27 154.24 225.95 227.43    167.84
 21  201.97 250.01 300.53 328.94    233.04 *
 22    8.15 154.89 195.61 329.82    146.75
 23  166.67 180.88 184.95 244.06    282.61 *
 24   20.01  97.10 127.25 145.74    234.27 *
 25   21.93  87.78 140.50 352.47    211.96 *
 26   56.04 131.40 170.81 176.36    239.68 *
 27   56.27  92.62 195.15 226.35    189.92 *
 28  140.57 182.19 235.62 337.88    162.69
 29   38.66 165.47 271.33 282.24    126.82
 30    6.75  75.73 214.59 299.78    138.86
 31   26.62  37.96  46.17 119.41    267.21 *
 32    0.09 193.25 195.85 236.54    193.16 *
 33   29.48  69.09 244.41 297.87    175.32
 34   53.99 128.53 163.51 253.58    160.41
 35   32.27 190.88 272.78 334.36    158.61
 36   74.75 144.66 166.27 177.18    257.57 *
 37   34.35  61.15 118.70 212.33    182.03 *
 38   35.25  98.26 159.79 333.94    174.15
 39   98.26 242.51 270.25 314.12    144.25
 40   11.14  32.36  92.39 116.18    254.96 *
 41   84.70 107.01 172.97 284.45    160.26
 42   16.18  91.66 122.62 173.67    202.50 *
 43   74.17 211.91 271.77 311.23    137.74
 44   29.05 119.17 195.46 334.04    138.58
 45   41.26 147.61 228.37 345.75    117.38
 46   53.73 125.18 223.28 332.44    109.16
 47   46.95  78.99 172.79 357.74    184.95 *
 48   10.40 124.34 197.16 332.26    135.10
 49  146.31 193.77 297.44 305.01    201.30 *
 50  122.33 242.07 259.88 358.84    123.49
 51   64.47 148.67 178.28 250.30    174.17
 52  152.25 194.73 195.54 293.28    218.97 *
 53   82.00 153.91 183.26 222.91    219.09 *
 54  133.39 176.34 245.09 319.18    174.21
 55   54.11 105.43 108.90 190.74    223.38 *
 56   80.37 130.85 210.43 315.35    125.02
 57   68.63 172.09 246.26 269.07    159.56
 58   58.23 221.02 281.57 290.80    162.79
 59   22.15  23.71  72.94 344.43    271.49 *
 60   43.04 136.66 166.89 285.55    118.66
 61   17.32  41.57  62.56 257.33    194.78 *
 62   78.02 168.48 191.89 201.96    236.06 *
 63  143.62 268.69 270.83 325.12    178.50
 64   31.88 228.45 256.69 268.56    196.56 *
 65    5.65  99.11 144.71 155.21    210.43 *
 66  150.34 250.60 288.93 354.73    155.60
 67   99.78 128.37 156.04 264.42    195.36 *
 68   43.76 125.22 232.54 340.19    107.65
 69   27.97  37.60  66.74 155.79    232.17 *
 70  177.98 195.13 345.34 350.27    187.71 *
 71   78.41 101.34 136.44 142.50    295.90 *
 72   49.94 181.21 186.21 347.53    161.32
 73  158.80 200.70 236.61 327.31    191.49 *
 74   23.22 249.48 252.19 272.19    226.26 *
 75   55.96  80.55 117.41 178.95    237.01 *
 76   18.09 186.49 272.53 282.42    168.39
 77  117.07 288.24 289.54 350.18    171.17
 78  149.99 242.94 315.28 325.85    184.15 *
 79   44.31 250.66 287.03 343.50    206.35 *
 80    5.87  59.12  60.39 144.58    221.29 *
 81   38.21  99.40 146.18 183.50    214.71 *
 82   67.60 179.27 231.50 305.68    121.92
 83  116.48 134.21 277.50 322.78    153.70
 84   78.48  84.97 160.92 316.22    155.30
 85  134.80 140.22 219.76 309.81    184.98 *
 86  118.91 186.19 210.94 335.36    143.55
 87   64.33  93.26  93.43 312.64    219.21 *
 88    0.90 124.90 267.24 302.72    142.34
 89  100.29 145.75 253.07 292.17    168.13
 90   28.07 147.83 157.56 267.04    121.03
 91  112.42 122.18 255.80 287.56    184.87 *
 92   54.63  87.53 213.48 344.25    130.77
 93   41.16 227.87 338.39 354.33    186.71 *
 94   88.26 206.95 215.59 324.93    123.34
 95   26.98 157.98 273.59 309.66    131.00
 96   88.44 136.30 143.01 189.57    258.88 *
 97   28.34  75.49  97.33 209.22    179.11
 98   40.12 234.86 322.36 324.12    194.74 *
 99   83.37 158.84 304.63 341.97    145.79
100  137.72 177.12 277.02 300.56    197.16 *
50
5036/10000=0.5036

DefDbl A-Z
Dim crlf$, r(4)


Private Sub Form_Load()
 Form1.Visible = True
 
 
 Text1.Text = ""
 crlf = Chr$(13) + Chr$(10)
 
 ntrials = 10000
 For trial = 1 To ntrials
   For i = 1 To 4
     r(i) = Rnd(1)
   Next
   ' numbers are 0 - 1, so units are one circumference
   Do
    done = 1
    For i = 1 To 3
      If r(i) > r(i + 1) Then h = r(i): r(i) = r(i + 1): r(i + 1) = h: done = 0
    Next
   Loop Until done
   mx = r(2) - r(1)
   For lwr = 2 To 3
     If r(lwr + 1) - r(lwr) > mx Then mx = r(lwr + 1) - r(lwr)
   Next
   lastarc = r(1) - r(4) + 1
   If lastarc > mx Then mx = lastarc
   If trial <= 100 Then
     Text1.Text = Text1.Text & mform(trial, "###") & " "
     For i = 1 To 4
     Text1.Text = Text1.Text & mform(360 * r(i), "###0.00")
     Next
     Text1.Text = Text1.Text & "   " & mform(mx * 360, "###0.00")
   End If
   If mx > 0.5 Then hits = hits + 1
   If trial <= 100 Then
     If mx > 0.5 Then Text1.Text = Text1.Text & " *"
     Text1.Text = Text1.Text & crlf
     If trial = 100 Then Text1.Text = Text1.Text & hits & crlf
   End If
   DoEvents
 Next
 Text1.Text = Text1.Text & hits & "/" & ntrials & "=" & hits / ntrials
 
 
 Text1.Text = Text1.Text & crlf & " done"
  
End Sub

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

Edited on March 29, 2017, 10:51 am
  Posted by Charlie on 2017-03-29 10:46:12

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