Between occurrences of C1, there can be any number of alternations between C2 and C3, starting of course with C2, but ending with either C2 or C3.
After a visit to C1, each successive city has a 3/4 chance of returning to C1. So the probability is 3/4 that the duration out of C1 is 1 day, and 3 / 4^2 that the duration is 2 days, etc.
So the expected length of time between C1 visits is
t = 3/4 + 2*3 / 4^2 + 3*3 / 4^3 + ...
4*t = 3 + 2*3 / 4 + 3*3 / 4^2 + 4*3 / 4^3 + ...
3*t = 3 + (6-3)/4 + (9-6) / 4^2 + (12-9) / 4/^3 + ...
t = 1 + 1/4 + 1/4^2 + 1/4^3 + ...
4*t = 4 + 1 + 1/4 + 1/4^2 + ...
= 4 + t
3*t = 4
t = 4/3
Mark spends 1 day out of every 7/3 days in C1 or 3/7 of his days.
Each excursion out of C1 starts with C2, and 3/4 of such an excursion ends right after that first such visit.
Finding the expected total duration at C2 during one outing from C1:
d = 1 + 1/4^2 + 1/4^4 + 1/4^6 + ...
16*d = 16 + d
15*d = 16
d = 16/15
Since such an excursion starts once every 7/3 days on average (i.e., 3/7 of days), Mark spends (3/7)*(16/15) of his days at C2. Reduced, that's 16/35.
That leaves 1 - 3/7 - 16/35 = 4/35 of Mark's days at C3.
Simulation verification:
DefDbl A-Z
Dim crlf$, city(3)
Private Sub Form_Load()
Form1.Visible = True
Text1.Text = ""
crlf = Chr$(13) + Chr$(10)
c = 1
For i = 1 To 10000
city(c) = city(c) + 1
Select Case c
Case 1
c = 2
Case 2, 3
If Rnd(1) < 0.75 Then
c = 1
Else
c = 5 - c
End If
End Select
Next
For i = 1 To 3
Text1.Text = Text1.Text & Str(city(i))
Next
Text1.Text = Text1.Text & crlf
For i = 1 To 3
Text1.Text = Text1.Text & Str(city(i) * 35 / 10000)
Next
Text1.Text = Text1.Text & crlf
Text1.Text = Text1.Text & crlf & " done"
End Sub
results in raw counts of
4282 4574 1144
for days out of 10,000 spent and each respective city.
When the fraction out of 10,000 is multiplied by 35 the numbers are:
14.987 16.009 4.004
in agreement with the number of 35ths found analytically.
The answers:
15/35 * 12,000 ~= 5143
16/35 * 12,000 ~= 5486
4/35 * 12,000 ~= 1371
Probabilistically verifying the ratios:
If, at generation zero, Mr. Markov started out, arbitrarily, in C1, and follow the probabilities in a Quantum multiverse (or actually 35 such quantum universes within the multiverse), in city 1. We follow these 35 Schrodinger's Marks and see what how many of them, including fractionated Marks are in each city, we find to the limit of our computer's precision that after 123 generations 15 are in city A, 16 in city B and 4 in city C. That is, the Markov chain also imitates the probability that on any given day Mark O.V. is in the given cities proportionally.
Days after day 0:
1 0 35 0
2 26.25 0 8.75
3 6.5625 28.4375 0
4 21.328125 6.5625 7.109375
5 10.25390625 23.10546875 1.640625
6 18.5595703125 10.6640625 5.7763671875
7 12.330322265625 20.003662109375 2.666015625
8 17.0022583007813 12.996826171875 5.00091552734375
9 13.4983062744141 18.2524871826172 3.24920654296875
10 16.1262702941895 14.3106079101563 4.5631217956543
11 14.1552972793579 17.267050743103 3.57765197753906
12 15.6335270404816 15.0497102737427 4.31676268577576
13 14.5248547196388 16.7127177119255 3.76242756843567
14 15.3563589602709 15.4654616117477 4.17817942798138
15 14.7327307797968 16.4009038172662 3.86636540293694
16 15.2004519151524 15.6993221305311 4.10022595431656
17 14.8496610636357 16.2255084037315 3.92483053263277
18 15.1127542022732 15.8308686967939 4.05637710093288
19 14.9154343482951 16.1268484775064 3.95771717419848
20 15.0634242387787 15.9048636418447 4.03171211937661
21 14.952431820916 16.0713522686228 3.97621591046118
22 15.035676134313 15.9464857985313 4.01783806715571
23 14.9732428992652 16.0401356511019 3.98662144963282
24 15.0200678255511 15.9698982616734 4.01003391277548
25 14.9849491308367 16.0225763037449 3.99247456541836
26 15.0112881518725 15.9830677721913 4.00564407593623
27 14.9915338860956 16.0126991708565 3.99576694304782
28 15.0063495854283 15.9904756218576 4.00317479271413
29 14.9952378109288 16.0071432836068 3.9976189054644
30 15.0035716418034 15.9946425372949 4.0017858209017
31 14.9973212686475 16.0040180970288 3.99866063432373
32 15.0020090485144 15.9969864272284 4.00100452425721
33 14.9984932136142 16.0022601795787 3.9992466068071
34 15.0011300897894 15.998304865316 4.00056504489468
35 14.999152432658 16.001271351013 3.99957621632899
36 15.0006356755065 15.9990464867402 4.00031783775326
37 14.9995232433701 16.0007151349448 3.99976162168506
38 15.0003575674724 15.9994636487914 4.00017878373621
39 14.9997318243957 16.0004022634065 3.99986591219785
40 15.0002011317032 15.9996983024452 4.00010056585162
41 14.9998491512226 16.0002262731661 3.99992457561129
42 15.0001131365831 15.9998302951254 4.00005656829153
43 14.9999151475627 16.000127278656 3.99995757378135
44 15.000063639328 15.999904541008 4.00003181966399
45 14.999952270504 16.000071594244 3.99997613525201
46 15.000035797122 15.999946304317 4.00001789856099
47 14.9999731521585 16.0000402717622 3.99998657607925
48 15.0000201358811 15.9999697961783 4.00001006794056
49 14.9999848980892 16.0000226528663 3.99999244904458
50 15.0000113264331 15.9999830103503 4.00000566321656
51 14.9999915051752 16.0000127422373 3.99999575258758
52 15.0000063711186 15.999990443322 4.00000318555932
53 14.999995221661 16.0000071675085 3.99999761083051
54 15.0000035837542 15.9999946243687 4.00000179187712
55 14.9999973121843 16.0000040317235 3.99999865609216
56 15.0000020158618 15.9999969762074 4.00000100793088
57 14.9999984881037 16.0000022678445 3.99999924405184
58 15.0000011339222 15.9999982991166 4.00000056696112
59 14.9999991495583 16.0000012756625 3.99999957477916
60 15.0000006378313 15.9999990432531 4.00000031891563
61 14.9999995216266 16.0000007175602 3.99999976081328
62 15.0000003587801 15.9999994618299 4.00000017939004
63 14.9999997309149 16.0000004036276 3.99999986545747
64 15.0000002018138 15.9999996972793 4.0000001009069
65 14.9999998486397 16.0000002270405 3.99999992431983
66 15.0000001135203 15.9999998297196 4.00000005676013
67 14.9999999148598 16.0000001277103 3.9999999574299
68 15.0000000638551 15.9999999042173 4.00000003192757
69 14.9999999521086 16.000000071837 3.99999997605432
70 15.0000000359185 15.9999999461222 4.00000001795926
71 14.9999999730611 16.0000000404083 3.99999998653056
72 15.0000000202042 15.9999999696938 4.00000001010208
73 14.9999999848469 16.0000000227297 3.99999999242344
74 15.0000000113648 15.9999999829527 4.00000000568242
75 14.9999999914764 16.0000000127855 3.99999999573818
76 15.0000000063927 15.9999999904109 4.00000000319636
77 14.9999999952055 16.0000000071918 3.99999999760273
78 15.0000000035959 15.9999999946061 4.00000000179795
79 14.9999999973031 16.0000000040454 3.99999999865154
80 15.0000000020227 15.999999996966 4.00000000101135
81 14.999999998483 16.0000000022755 3.99999999924149
82 15.0000000011378 15.9999999982934 4.00000000056888
83 14.9999999991467 16.00000000128 3.99999999957334
84 15.00000000064 15.99999999904 4.00000000032
85 14.99999999952 16.00000000072 3.99999999976
86 15.00000000036 15.99999999946 4.00000000018
87 14.99999999973 16.000000000405 3.999999999865
88 15.0000000002025 15.9999999996963 4.00000000010125
89 14.9999999998481 16.0000000002278 3.99999999992406
90 15.0000000001139 15.9999999998291 4.00000000005695
91 14.9999999999146 16.0000000001281 3.99999999995729
92 15.0000000000641 15.9999999999039 4.00000000003204
93 14.9999999999519 16.0000000000721 3.99999999997597
94 15.000000000036 15.9999999999459 4.00000000001802
95 14.999999999973 16.0000000000405 3.99999999998649
96 15.0000000000203 15.9999999999696 4.00000000001014
97 14.9999999999848 16.0000000000228 3.9999999999924
98 15.0000000000114 15.9999999999829 4.0000000000057
99 14.9999999999915 16.0000000000128 3.99999999999572
100 15.0000000000064 15.9999999999904 4.00000000000321
101 14.9999999999952 16.0000000000072 3.9999999999976
102 15.0000000000036 15.9999999999946 4.00000000000181
103 14.9999999999973 16.0000000000041 3.99999999999865
104 15.000000000002 15.999999999997 4.00000000000102
105 14.9999999999985 16.0000000000023 3.99999999999924
106 15.0000000000011 15.9999999999983 4.00000000000057
107 14.9999999999991 16.0000000000013 3.99999999999957
108 15.0000000000006 15.999999999999 4.00000000000032
109 14.9999999999995 16.0000000000007 3.99999999999976
110 15.0000000000004 15.9999999999995 4.00000000000018
111 14.9999999999997 16.0000000000004 3.99999999999987
112 15.0000000000002 15.9999999999997 4.0000000000001
113 14.9999999999999 16.0000000000002 3.99999999999992
114 15.0000000000001 15.9999999999998 4.00000000000006
115 14.9999999999999 16.0000000000001 3.99999999999996
116 15.0000000000001 15.9999999999999 4.00000000000003
117 15 16.0000000000001 3.99999999999998
118 15 15.9999999999999 4.00000000000002
119 15 16 3.99999999999999
120 15 16 4.00000000000001
121 15 16 3.99999999999999
122 15 16 4.00000000000001
123 15 16 4
124 15 16 4
125 15 16 4
126 15 16 4
127 15 16 4
128 15 16 4
129 15 16 4
130 15 16 4
131 15 16 4
132 15 16 4
133 15 16 4
DefDbl A-Z
Dim crlf$, city(1, 3)
Private Sub Form_Load()
Form1.Visible = True
Text1.Text = ""
crlf = Chr$(13) + Chr$(10)
city(0, 1) = 1
For i = 1 To 150
DoEvents
city(1, 2) = city(0, 1) + city(0, 3) / 4
city(1, 3) = city(0, 2) / 4
city(1, 1) = 3 * (city(0, 2) + city(0, 3)) / 4
Text1.Text = Text1.Text & i & " "
For j = 1 To 3
city(0, j) = city(1, j)
Text1.Text = Text1.Text & Str(city(0, j) * 35)
Next
Text1.Text = Text1.Text & crlf
Next
Text1.Text = Text1.Text & crlf
Text1.Text = Text1.Text & crlf & " done"
End Sub
|
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Posted by Charlie
on 2016-05-26 14:22:24 |
Analytic + Simulation + Probabilistic
