2. UNR, MATH/STAT 352, Spring 2007

5. Predictable, certain, deterministic Unpredictable, uncertain Gravity acceleration g ~ 9.8 m/s 2 h = ½ gt 2 E = mgh=mv 2 /2 A black hole UNR, MATH/STAT 352, Spring 2007

6. Coin as a rigid body obeys classical gravitation laws, and its fall is deterministic & predictable… … it is well described by physical laws Coin as a gambling tool (also obeying gravitation laws) is unpredictable… TAIL? HEAD? h = ½ gt 2 … so its fall is better described by probabilistic laws UNR, MATH/STAT 352, Spring 2007

7. TAIL? HEAD? h = ½ gt 2 Level of detail may turn deterministic phenomena into random and vice versa… UNR, MATH/STAT 352, Spring 2007

8. Life time of a device (computer chip) Time Failure Shooting a target with a fixed riffle micro-defects, change in temperature condition, transportation, storage, energy power,… Origin of randomness: defects in bullets masses, density inhomogeneities, changing atmospheric-conditions, … UNR, MATH/STAT 352, Spring 2007

10. Max Born (December 11, 1882 – January 5, 1970) Angular momentum Energy Electron position measurements High probability Low probability UNR, MATH/STAT 352, Spring 2007

11. 1954 Nobel Prize in Physics "for his fundamental research in quantum mechanics, especially for his statistical interpretation of the wavefunction" Max Born (December 11, 1882 – January 5, 1970) UNR, MATH/STAT 352, Spring 2007

12. 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

13. 1: Single out a set of primary factors 2: Determine their interrelationships 3: Choose appropriate mathematical apparatus 4: Determine outcome given primary factors 5: Add new primary factors UNR, MATH/STAT 352, Spring 2007

14. 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

15. 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

16. Da Vinci (1452-1519) UNR, MATH/STAT 352, Spring 2007

17. Weather = Dynamics of atmosphere = Turbulence ! UNR, MATH/STAT 352, Spring 2007

18. Hurricane Floyd, 1999

19. Andrei Kolmogorov (1903-1987): A founder of modern theory of probabilities (1933) Da Vinci (1452-1519) Claude Louise Mary Henry Navier (1821) George Gabriel Stokes Turbulence Phenomenon Classical (deterministic) approach Probabilistic approach Navier-Stokes EQs UNR, MATH/STAT 352, Spring 2007

21. Possible outcomes of coin tossing: HE AD TAI L RI B LOS T Relevant, interesting, most probable outcomes: HE AD TAI L Model: { H, T } UNR, MATH/STAT 352, Spring 2007

22. Observations Model Probability (a fair coin will show about 50% of tails) Statistics (a coin that shows 90 tails out of 100 throws is probably not fair) UNR, MATH/STAT 352, Spring 2007

23. … 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

24. Q1: What % of bullets will on average hit the target? Q2: How many bullets should be spent to hit the target almost surely? Q3: 14 bullets out of 100 hit the target: Was it riffle A? A: 75%A: from 60% to 90% samplepopulation A: 3 bullets Riffle A hit the target with 75 bullets out of 100 A: Most probably not Point estimation Interval estimation Hypothesis testing UNR, MATH/STAT 352, Spring 2007

26. % newborn boys (about 51%) 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 Car accidents Street traumatism Crime rates UNR, MATH/STAT 352, Spring 2007

27. 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

28. $$$ = ??? $$$ = !!! UNR, MATH/STAT 352, Spring 2007

29. Luggage = ??? UNR, MATH/STAT 352, Spring 2007

30. 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