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01. Sample Space

郭俊利 2009/02/27. 01. Sample Space. 1.1 ~ 1.5. Outline. Sample space Probability axioms Conditional probability Independence. Introduction. What is probability? Time Frequency Space Area Examples Weather forecast ^^ Cancer prediction ^^ Lottery > ” <. Sets. Sample space

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01. Sample Space

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  1. 郭俊利 2009/02/27 01. Sample Space

  2. 1.1 ~ 1.5 Outline • Sample space • Probability axioms • Conditional probability • Independence

  3. Introduction • What is probability? • Time • Frequency • Space • Area • Examples • Weather forecast ^^ • Cancer prediction ^^ • Lottery >”<

  4. Sets • Sample space • List of all possible outcomes • S1 = {H, T} (H = head; T = tail) • S2 = { (H, H), (H, T), (T, H), (T, T) } • Event • A subset of the sample space

  5. Basic Laws • Axioms: 1. 0 ≦ P(A) ≦ 1 2. P(S) = P(universe) = 1 3. If A∩B = Ø, then P(A∪B) = P(A) + P(B) 4. If A∩B ≠ Ø, …

  6. Real Laws • P(A∪B) = P(A) + P(B) – P(A∩B) = P(A) + P(AC∩B) • P(A∪B∪C) = P(A) + P(B) + P(C) – P(A∩B) – P(B∩C) – P(C∩A) + P(A∩B ∩C) • P(A∪B∪C) = P(A) + P(AC∩B) + P(AC∩BC∩C)

  7. Example 1 • Bonferroni’s inequality • P(A∩B) ≧ P(A) + P(B) – 1 • P(A1∩A2∩…∩An) ≧ P(A1) + P(A2) + … + P(An) – (n – 1)

  8. P(A∩B) P(B) P(A|B) = Example 2 • Given that the two dice land on different numbers, find the conditional probability that at least one die roll is a 6.

  9. P(A∩B) P(B) P(A|B) = Multiplication Rule • P(A∩B) = P(B) P(A|B) • P(A∩B∩C) = P(A) P(B|A) P(C|A∩B)

  10. Example 3

  11. Independence • P(A∩B) = P(B) P(A|B) • If A is independent of B, P(A) = P(A|B) P(A∩B) = P(B) P(A) • If A is disjoint of B, then A is independent of B?

  12. Example 4 36 2 3 2 3 1 4 5

  13. Example 5 • You enter a chess tournament where your probability of winning a game is 0.3 against half the players, 0.4 against a quarter of the players, and 0.5 against the remaining quarter of the players. You play a game against a randomly chosen opponent. • What is the probability of winning? • Suppose that you win. What is the probability that you had an opponent of 3rd type?

  14. P(A∩B∩C) P(C) P(C) P(B|C) P(A|B∩C) P(C) Conditional Independence • P(A∩B | C) = P(A|C) P(B|C) P(A|B∩C) = P(A|C) P(A∩B | C) = = = P(B|C) P(A|B∩C)

  15. Example 6 • H1 and H2 are independent, but not conditionally Independent H1 = {1st toss is a head} H2 = {2nd toss is a head} D = {the two tosses have different results} P(H1|D) = ½; P(H2|D) = ½; P(H1∩H2 | D) = 0

  16. Example 7 • Are H1, H2 and D independent? H1 = {1st toss is a head} H2 = {2nd toss is a head} D = {the two tosses have different results} H1 H2 D

  17. Example 8 • Let A and B be independent. • Are A and BC independent? • Are AC and BC independent?

  18. Example 9 • Let A, B , C be independent. • Prove that A and B are conditionally Independent given C. P(A∩B | C) = P(A|C) P(B|C)

  19. 0.8 E 0.9 C 0.9 0.85 0.95 B A F 0.95 0.75 D Example 10 (1/2)

  20. Example 10 (2/2) 0.8 E 0.9 C 0.9 0.85 0.95 B A F 0.95 0.75 D

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