1. P robability 1 01. Sample Space 郭俊利 2009/02/27

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

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

4. Probability 4 Sets Sample space List of all possible outcomes S 1 = {H, T} (H = head; T = tail) S 2 = { (H, H), (H, T), (T, H), (T, T) } Event A subset of the sample space

5. Probability 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. Probability 6 Real Laws P(A ∪ B) = P(A) + P(B) – P(A ∩ B) = P(A) + P(A C ∩ 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(A C ∩ B) + P(A C ∩ B C ∩ C)

7. Probability 7 Example 1 Bonferroni ’ s inequality P(A ∩ B) ≧ P(A) + P(B) – 1 P(A 1 ∩ A 2 ∩…∩ A n ) ≧ P(A 1 ) + P(A 2 ) + … + P(A n ) – (n – 1)

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

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

10. Probability 10 Example 3

11. Probability 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. Probability 12 Example 4 36 23 1 2 3 4 5

13. Probability 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 3 rd type?

14. Probability 14 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) P(A∩B∩C) P(C) P(C) P(B|C) P(A|B∩C) P(C) =

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

16. Probability 16 Example 7 H 1 = {1 st toss is a head} H 2 = {2 nd toss is a head} D = {the two tosses have different results} H1H1 H2H2 D Are H 1, H 2 and D independent?

17. Probability 17 Example 8 Let A and B be independent. Are A and B C independent? Are A C and B C independent?

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

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