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).
(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
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Posted by Charlie
on 2017-03-29 10:46:12 |
re: Corrected solution
