UNR, MATH/STAT 352, Spring 2007
UNR, MATH/STAT 352, Spring 2007
UNR, MATH/STAT 352, Spring 2007
Predictable, certain,
deterministic
Unpredictable, uncertain
Gravity acceleration
g ~ 9.8 m/s2
h = ½ gt2
E = mgh=mv2/2
UNR, MATH/STAT 352, Spring 2007
Coin as a rigid body obeys classical
gravitation laws, and its fall is
deterministic & predictable…
Coin as a gambling tool (also obeying
gravitation laws) is unpredictable…
HEAD?
h = ½ gt2
TAIL?
… it is well described by
… so its fall is better described by
physical laws
UNR, MATH/STAT 352, Spring 2007
probabilistic laws
Level of detail may turn deterministic phenomena into random and
vice versa…
HEAD?
h = ½ gt2
TAIL?
UNR, MATH/STAT 352, Spring 2007
Life time of a device (computer chip)
Failure
Time
Origin of randomness:
micro-defects, change in temperature condition, transportation,
storage, energy power,…
Shooting a target with a fixed riffle
Origin of randomness:
defects in bullets masses, density inhomogeneities,
changing atmospheric-conditions, …
UNR, MATH/STAT 352, Spring 2007
UNR, MATH/STAT 352, Spring 2007
Electron position
measurements
Max Born
Angular momentum
Energy
Low probability
High probability
(December 11, 1882 – January 5, 1970)
UNR, MATH/STAT 352, Spring 2007
Max Born
1954 Nobel Prize in Physics
"for his fundamental research
in quantum mechanics, especially for his
statistical interpretation of the wavefunction"
(December 11, 1882 – January 5, 1970)
UNR, MATH/STAT 352, Spring 2007
Albert Einstein
(March 14, 1879 – April 18, 1955)
Quantum mechanics is certainly imposing. But an inner
voice tells me it is not yet the real thing. The theory says
a lot, but does not really bring us any closer to the secret
of the Old One. I, at any rate, am convinced that He does
not throw dice.
(Letter to Max Born,1926)
UNR, MATH/STAT 352, Spring 2007
1: Single out a set
of primary factors
2: Determine their
interrelationships
5: Add new primary
factors
3: Choose appropriate
mathematical apparatus
4: Determine outcome
given primary factors
UNR, MATH/STAT 352, Spring 2007
20th century has witnessed the Science revolution that
demonstrated a limited power of the classical approach, its
inability to predict evolution of many natural phenomena.
This happens because many processes crucially depend on
a countless number of uncontrolled parameters and their interplay.
Insignificant and undetectable changes in initial conditions
may (and do) lead to different result of an entire experiment.
Say, try to figure out what controls the position (head vs. tail)
of a coin.
UNR, MATH/STAT 352, Spring 2007
There should be a principal difference in methods
and approaches applied to description of a small
number of well controlled primary factors and
a large number of uncontrollable secondary factors.
TP & S are among sciences targeted at solving this
problem (other are theory of deterministic chaos,
theory of complexity, etc.)
UNR, MATH/STAT 352, Spring 2007
Da Vinci (1452-1519)
UNR, MATH/STAT 352, Spring 2007
Weather
= Dynamics of atmosphere
= Turbulence !
UNR, MATH/STAT 352, Spring 2007
Hurricane Floyd, 1999
Phenomenon
Turbulence
Probabilistic approach
Da Vinci (1452-1519)
Classical (deterministic) approach
George Gabriel
Stokes
Claude Louise
Mary Henry
Navier (1821)
Andrei Kolmogorov (1903-1987):
A founder of modern theory of
probabilities (1933)
Navier-Stokes EQs
UNR, MATH/STAT 352, Spring 2007
UNR, MATH/STAT 352, Spring 2007
Possible outcomes of coin tossing:
HEA TAIL RIB LOST
D
Relevant, interesting, most probable outcomes:
HEA TAIL
D
Model: { H, T }
UNR, MATH/STAT 352, Spring 2007
Probability
(a fair coin will show about 50% of tails)
Model
Ek ( g )   g ( x ) Fk ( x )
TABLE 2.1.1 Data on Male Heart Attack Patients
S YS DIAOUTVOL
VOL
OCCLU
S TEN
TIME
COME
AGE
S MO KE
36
131
0
0
143
0
49
2
74
155
37
63
143
0
54
2
52
137
33
47
16
2
56
2
165
329
33
30
143
0
42
2
47
95
0
100
143
0
46
2
TABLE 2.1.1
Patients
124Data on Male
170Heart Attack 77
23
143
0
57
2
86
215
7
40
0
51
2
SYSDIAOUT-50
37
132
40 TIME COME10 AGE
5a SURG
56
2
ID
EJEC
VOL
VOL OCCLU STEN
SMOKE 9 BETA CHOL
65
163
45
2
390
72
36
131
0
0 0 143
0 40 49
2142 2
59 0
0
52
140
46
2
279
52
74
155
37
63 0 143
0 10 54
2142 2
68 0
1
TABLE
2.1.1
Da
ta
on
Male
Heart
Atta
ck
Patients
117
164
50
0
142
0
48
2
391
62
52
137
33
47
16
2
56
2
2S YS - 52
0
DIAID 42
EJEC
OC CLU
S TEN
69
133
54
2 TIME
201
50
165
329
33
30 0 143
0 27
2142 2VOL
39 0 VOL
0
390
72
36
131
0
0
143
54
133
39
2 143
279 46
52
37
63
202
50
47
95
0
10030
143
0 13
2142 2 74 74 0 155
1
391
62
52
137
33
47
16
67
135
49
2 143
69
27
124
170
77
2337
143
0 63
2141 2 165 NA0 329
2
201 57
50
33
30
202
50
47
95
0
100
143
65
138
58
2 143
310
60
86
215
7
50 0
40
0 3369 51
2140 2 124 58 0 170
0
27
77
23
310
60
86
215
7
50
40
184
221
50
2
392
72
37
132
40
1057
9
5 13
2 5 2 37
75 1 132
0
392 56
72
40
10
9
311
60
65
163
0
40
142
88
140
58
2 142
311
60
65
163
0
4037
142
0 47
45
2118 2 52
72 5 140
0
393
63
0
10
70 46
29
50
0
106
193
140 2 117 90 0 164
47
1 142
393
63
52
140
0
1033
142
0 43
2
0
203
48
69
133
0
27
142
85
150
51
2 142
394 48
59
30
13
70
29
117
164
50
0 0 142
0 50
2 23 2 54 72 5 133
0
204
50
67
135
37
63
141
59
149
7
37
139
0
43
2
203
48
69
133
0
27
142
0 280 54
2
2 65 NA
0
53
138
0
33
140
55
17
184
221
57
13
103
168
55
2 1185
394
59
54
133
30
1347
142
0 4379 39
2100 1 88 NA1 140
0
37
37
47
205
45
106
193
33
43
140
53
124
58
2
204
50
67
135
37
63 0 141
0 57
2140 2 85 86 0 150
2
206 49
43
0
50
23
312
60
59
149
7
37
139
68
121
55
2 100
280
53
65
138
0
3327
140
0 60
58
2139 1 103
49 0 168
0
80
38
47
43
281 50
57
53 70 0
124
0
57
53
109
41
2 140
55
17
184
221
57
13 0
5
1 77
2139 2 68
2
207
44
121
27
60
139
282 58
51
0
77
58
157
51
2 139
79
37
88
140
37
47 0 118
5 73
2139 2 53 NA0 109
0
396
63
58
157
0
73
139
81
157
49
2 139
208 47
49
13
13
205
45
106
193
33
4313
140
0 13
1139 1 81 38 0 157
1
209
48
58
112
0
0
72
58
112
0 51
56
2 138
206
43
85
150
0
50 0
23
5 283
2 72 2 71 61 1 167
0
58
27
0
210
42
92
159
0
0
139
71
167
0 43
45
2 138
312
60
59
149
7
3727
139
0 397
2138 1 50
56 0 156
0
68
0
100
211
43
146
259
47
33
3
92
159
0
0
139
0
57
2
80
38
103
168
47
43
100
1 398 55
2
2 43 62
1
67
130
0
70
138
284 58
52
70 93 0
146
0
23
50
156
51
2 137
281
57
53
124
0
57 0 140
0100
2138 1 73
0
399
63
195
27
0
136
285 55
54
33
23
146
259
56
2 137
207
44
68
121
27
6047
139
0 33
2 3 2 62 63 1 133
1
71
37
93
148
47
0
137
43
130
49
2 136
286 41
51
43
7
282
51
53
109
0
77 0 139
0 70
2138 2 65 45 0 133
4
212
42
95
163
40
10
70
146
47
1 109
396
63
58
157
0
73 0 139
0 23
2137 2 49 60 0 144
0
400 51
66
10
50
65
54
66
145
7
40
136
73
195
081 49
36
1 136
208
49
81
157
13
1327
139
0 287
2136 2 144
60 0 237
0
39
13
87
813
63
52
141
0
47
43
62
133
33
38
2
209
48
58
112
0
0
72
1 23
56
2137 2 219
57 0 314
0
68
30
33
45
76
59
94
0
0
93
148
47
0 45
59
2 135
283
58
71
167
27
0
138
0 288
2137 1 39
46 0 117
0
407
67
39
0
73
53
g d ata cod e).
65
133
43
54
2
210
42
92
159
0
0
139 NA 0= No7t A vai57lab le(missin2136
2
58 0
0
95
163
57
2
397
68
50
156
0
10040
138
0 10 51
2109 1
NA3
0
49
144
52
2
211
43
146
259
47
3310
3
1 50 56
2 65 2
70 1
0
66
145
47
2
398
67
43
130
0
70 7 138
0 40 49
2136 2
NA0
3
144
237
39
2
284
52
70
146
0
2313
137
0 87 47
1136 2
NA0
0
52
141
48
2
399
63
73
195
27
0 0 136
0 47 36
1 43 1
61 3
0
219
314
53
1
285
54
62
133
33
2333
137
0 45 38
2 76 2
NA1
0
39
94
47
1
71
37
93
148
47
0 0 137
0 0 59
2135 2
NA0
0
39
117
57
2
286
51
65
133
43
7 0 136
0 73 54
2 53 2
NA1
0
Not A vai lable(missing data
212code). 42
95
163
40
10
109
3
57
2
2
NA
4
400
66
49
144
10
50
65
1
52
2
2
55
0
287
54
66
145
7
40
136
0
47
2
2
62
0
81
39
144
237
13
87
136
0
39
2
2
56
3
813
63
52
141
0
47
43
3
48
2
2
NA
0
68
30
219
314
33
45
76
1
53
1
2
NA
0
288
59
39
94
0
0
135
0
47
1
2
63
0
407
67
39
117
0
73
53
1
57
2
2
62
2
ID
390
279
391
201
202
69
310
392
311
393
70
203
394
204
280
55
79
205
206
312
80
281
207
282
396
208
209
283
210
397
211
398
284
399
285
71
286
212
400
287
81
813
68
288
407
a
NA =
EJEC
72
52
62
50
50
27
60
72
60
63
29
48
59
50
53
17
37
45
43
60
38
57
44
51
63
49
48
58
42
68
43
67
52
63
54
37
51
42
66
54
39
63
30
59
67
Observations
a
a
NA = Not Available(missing data code).
Ek ( g )   g ( x ) Fk ( x )
Ek ( g )   g ( x ) Fk ( x )
Statistics
(a coin that shows 90 tails out of 100 throws is probably not fair)
UNR, MATH/STAT 352, Spring 2007
BETA
2
2
2
2
2
2
2
2
2
2
2
OUTCOME
2
0
1
0
2
2
0
0
1
0
0
2
5
0
2
0
0
1
0
2
0
0
1
0
1
2
5
0
1
5
0
2
1
0
2
0
0
2
0
2
0
1
2
0
0
1
0
1
2
0
0
1
0
0
2
0
2
0
3
2
1
0
1
0
3
2
1
0
2
1
2
2
2
2
2
2
2
2
2
CHO La
S URG
59
0
68
1
52
0
39
0
74
1
NA
2
58
0
75
0
72
0
90
0
72
0
AGE
NA S MO KE 0 BETA
49
2
2
NA
0 2
54
2
56
2
2
86
2 2
42
2
46
2
49
0 22
57
2
51
2
70
2 22
56
2
45
2
NA
0 22
46
2
48
2
38
1 2
54
2
2
61
0 1
39
2
49
2
2
56
0
58
2
1
50
2
62
1 22
58
2
47
1
93
0 12
51
2
43
2
63
1 21
55
2
58
2
45
4 1
55
2
2
41
2
60
0 2
51
2
2
60
0 2
49
2
56
2
57
0 12
45
2
57
2
46
0 12
51
2
56
2
58
0 22
49
2
47
1
NA
0 2
36
1
1
38
2
70
0 2
59
2
2
NA
3 2
54
2
57
2
NA
0 22
52
2
47
2
61
0 22
39
2
48
2
NA
0 22
53
1
47
1
2
NA
0
57
2
2
NA
0
NA
4
55
0
62
0
56
3
NA
0
NA
0
63
0
62
2
CHO La
59
68
52
39
74
NA
58
75
72
90
72
NA
NA
86
49
70
NA
38
61
56
62
93
63
45
60
60
57
46
58
NA
70
NA
NA
61
NA
NA
NA
NA
55
62
56
NA
NA
63
62
S URG
0
1
0
0
1
2
0
0
0
0
0
0
0
2
0
2
0
1
0
0
1
0
1
4
0
0
0
0
0
0
0
3
0
0
0
0
0
4
0
0
3
0
0
0
2
… work when we can describe possible outcomes
of experiment, but can not predict its specific outcome
… deal with appropriate mathematical model of
a physical phenomenon, not with phenomenon itself
UNR, MATH/STAT 352, Spring 2007
sample
population
Riffle A hit the target with
75 bullets out of 100
Q1: What % of bullets will on average hit the target?
A: 75%
A: from 60% to 90%
Q2: How many bullets should be spent to hit the target
almost surely?
A: 3 bullets
Q3: 14 bullets out of 100 hit the target: Was it riffle A?
A: Most probably not
UNR, MATH/STAT 352, Spring 2007
UNR, MATH/STAT 352, Spring 2007
TP & S use the phenomenon of stability of frequencies:
Observing a large number of uniform random events,
we often detect amazing regularities
# tails
# bets

1/2
% quality goods
% newborn boys
(about 51%)
UNR, MATH/STAT 352, Spring 2007
Earthquakes: the most unpredictable
Natural disaster
Gutenberg-Richter law
Prediction:
In 2007 there will be about
1000 EQs with magnitude 5, and
10 EQs with M7
UNR, MATH/STAT 352, Spring 2007
$$$ = ???
$$$ = !!!
UNR, MATH/STAT 352, Spring 2007
Luggage = ???
UNR, MATH/STAT 352, Spring 2007
TP&S do not cancel randomness, unpredictability of individual
experiment. They allow to predict in some approximation an
average result of a mass of uniform random phenomena.
The goal of TP&S is to overpass a difficult (and often practically
impossible) exploration of a single random phenomena and jump
to collective behavior.
Probabilistic predictions are different from exact statement
of what, when, and where to expect. They establish boundary
within which, with a high degree of reliability one will observe
interesting phenomenon. The larger the number of individual
events, the sharper the possible probabilistic prediction.
UNR, MATH/STAT 352, Spring 2007