Dept. Computer Science, Korea Univ. Intelligent Information System Lab. 1 In-Jeong Chung Intelligent Information System lab. Department of Computer Science Korea University Probability & Statistics #2

Dept. Computer Science, Korea Univ. Intelligent Information System Lab. 1.4 Independent Events Def) A, B, C are mutually independent iff ■1) A, B, C are pairwise independent i.e.  P(A ∩ B) = P(A) P(B),  P(A ∩ C) = P(A) P(C),  P(B ∩ C) = P(B) P(C), ■2) P(A ∩ B ∩ C) = P(A) P(B) P(C) ■e.g.  ex : not mutually independent, but pairwise independent 2

Dept. Computer Science, Korea Univ. Intelligent Information System Lab. 1.4 Independent Events ■e.g. (p.30 ex )  Event A i : component C i fails (i = 1, 2, 3)  Probability that system would work correctly = probability that system does not fail = 3 C1C1 C2C2 C3C3

Dept. Computer Science, Korea Univ. Intelligent Information System Lab. 1.4 Independent Events ■e.g. (p.31 ex )  5 consecutive days  “instant winner” lottery ticket  Probability of winning = 1/5 on ∀ day  Independent trials  Generally, probability of 2 winning tickets & 3 losing tickets 4

Dept. Computer Science, Korea Univ. Intelligent Information System Lab. 1.5 Bayes’s Theorem Thomas Bayes (1761) Conditional probability ■Motivating e.g. traffic accident  40% of traffic accident : speed ↑event(E)  30% of traffic accident : C 2 H 5 OHevent(A)  If C 2 H 5 OH is involved,60% of speed ↑  o.w10% of speed ↑  traffic accident involves speed ↑ ■Probability that C 2 H 5 OH is involved 5

Dept. Computer Science, Korea Univ. Intelligent Information System Lab. 1.5 Bayes’s Theorem 6 i.e. If speed ↑ was involved in an traffic accident, 72% chance of C 2 H 5 OH involvement Speed ↑ & C 2 H 5 OH not involved Speed ↑ & C 2 H 5 OH involved

Dept. Computer Science, Korea Univ. Intelligent Information System Lab. 1.5 Bayes’s Theorem Def) Let ■B 1,…,B k : partition of the sample space S ■A : event of S i = 1, 2, …, k Bayes’ thm ■Probability of a particular B i, given that an event A has occured 7

Dept. Computer Science, Korea Univ. Intelligent Information System Lab. 1.5 Bayes’s Theorem ■e.g. (p.35 ex )  A spring in selected at random, probability that it is defective  If the selected spring is found to be defective,  Probability that it was produced by m/c Ⅲ 8 m/c Ⅰ m/c Ⅱ m/c Ⅲ Defective rate2%1%3% Production portion35%25%40%

Dept. Computer Science, Korea Univ. Intelligent Information System Lab. 1.5 Bayes’s Theorem ■e.g. pap smear : used to detect cervical cancer ■Cervical cancer : 8/100,000 9

Dept. Computer Science, Korea Univ. Intelligent Information System Lab. 1.5 Bayes’s Theorem Bayes thm ■P(patient : cancer | test result = +) ■i.e. for ∀ 10 6 positive pap smears, only 354 represents true cases of cervical cancer 10

Dept. Computer Science, Korea Univ. Intelligent Information System Lab. Homework HW ■[1.1] 4, 6, 10 ■[1.2] 10 ■[1.3] 4, 10, 12 ■[1.4] 2, 6, 8, 10, 12 ■[1.5] 2, 6, 8 11