Introduction to Probability Theory ‧1‧

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1 Introduction to Probability Theory ‧1‧
- Preliminaries for Randomized Algorithms Speaker: Chuang-Chieh Lin Advisor: Professor Maw-Shang Chang National Chung Cheng University Dept. CSIE, Computation Theory Laboratory January 4, 2006

2 Computation Theory Lab., Dept. CSIE, CCU, Taiwan
For convenience, we omit the contents of set theory concepts, counting techniques, and the discussion of sample spaces and probability measures. Computation Theory Lab., Dept. CSIE, CCU, Taiwan

3 Computation Theory Lab., Dept. CSIE, CCU, Taiwan
Outline Chapter 1: Probability Some useful theorems and principles Conditional probability Theorem of total probability Bayes’ theorem Independent events Independent trails Computation Theory Lab., Dept. CSIE, CCU, Taiwan

4 Kolmogorov axioms (柯莫格洛夫公理)
1. For any event A  S, P(A)  0. 2. P(S) = 1. 3. If A1, A2, … are mutually exclusive events, then P(A1A2 …) = P(A1) + P(A2) + … . Computation Theory Lab., Dept. CSIE, CCU, Taiwan

5 Computation Theory Lab., Dept. CSIE, CCU, Taiwan
Useful theorems P() = 0 for any experiment. For any event A  S, P(A) = 1  P(A). If A  S, B  S are any two events, then P(AB) = P(A) + P(B)  P(AB). If A  B, then P(A)  P(B). _ Computation Theory Lab., Dept. CSIE, CCU, Taiwan

6 Conditional probability
In an experiment with sample space S, let B be any event such that P(B) > 0. Then the conditional probability of A occurring, given that B has occurred, is for any A  S. Computation Theory Lab., Dept. CSIE, CCU, Taiwan

7 Computation Theory Lab., Dept. CSIE, CCU, Taiwan
假設你在超市買牛奶，總共有 40 盒供你選，但其中有 10 盒已腐壞（外表看不出來）。假設你買了兩盒牛奶，兩盒都是好的機率是多少？ 令 A 事件代表你選的第一盒是好的，令 B 事件代表你選的第二盒是好的。 Computation Theory Lab., Dept. CSIE, CCU, Taiwan

8 Theorem of total probability
If A1, A2,…, An is a partition of S, and B is any event, then B A1 A2 A3 A4 A5 A6 A7 Computation Theory Lab., Dept. CSIE, CCU, Taiwan

9 Computation Theory Lab., Dept. CSIE, CCU, Taiwan
From the theorem of total probability, and granted that A1, A2,…, An is a partition of S, we have This result is known as Bayes’ theorem (貝式定理). Computation Theory Lab., Dept. CSIE, CCU, Taiwan

10 Computation Theory Lab., Dept. CSIE, CCU, Taiwan
假設被選中參加一項刑案審判的陪審團，不論被告有罪或無罪，都有 95% 的機會做出正確判決。另外還假設當地警方執法非常嚴格，在被審判的人當中，有 99% 事實上是有罪的。若已知陪審團判某被告無罪，則該名被告真的是無罪的機率是多少？ 令 A1 代表被告有罪之事件，則 代表他無罪的事件。 再令 B 代表被告獲判無罪的事件。 我們要算的是 P(A2 | B)。 Computation Theory Lab., Dept. CSIE, CCU, Taiwan

11 Computation Theory Lab., Dept. CSIE, CCU, Taiwan
在審判前，這名被告無罪的機率為 0.01，在他獲判無罪之後，這個值增加到了0.161。 Computation Theory Lab., Dept. CSIE, CCU, Taiwan

12 Computation Theory Lab., Dept. CSIE, CCU, Taiwan
Independent events If A  S and B  S are any two events with nonzero probabilities, A and B are called independent if and only if P(A  B) = P(A)P(B) That is, P(A) = P(A | B) and P(B) = P(B | A). Computation Theory Lab., Dept. CSIE, CCU, Taiwan

13 Computation Theory Lab., Dept. CSIE, CCU, Taiwan
Independent trials An experiment is said to consist of n independent trials if and only if S = T1  T2  …  Tn. For every (x1, x2, …, xn)  S, P({(x1, x2, …, xn)}) = P1({x1})P2({x2}) … Pn({xn}), where Pi({xi}) is the probability of xi  Ti occurring on trial i. Computation Theory Lab., Dept. CSIE, CCU, Taiwan

14 Computation Theory Lab., Dept. CSIE, CCU, Taiwan
假設力穎在修演算法這門課時，某天老師給了一個隨堂考，有 4 題是非題和 3 題選擇題(每題有 3 個選項)。每一小題只有一個對的答案。因為力穎都不會，因此每一題的答案都用猜的。假設力穎至少得答對六題才算及格。 我們可用七個獨立試驗(independent trials)來描述這個事件。 設 T = {r, w} (i.e., right answer and wrong answer)，則sample space S = T T T T T T T 令 A 事件代表他隨堂考及格。 Computation Theory Lab., Dept. CSIE, CCU, Taiwan

15 Computation Theory Lab., Dept. CSIE, CCU, Taiwan
Pi({r}) = 1/2 if i = 1, 2, 3, 4, else, Pi({r}) = 1/3. 則 P(A) = 所以力穎的狗運要好才行。 Computation Theory Lab., Dept. CSIE, CCU, Taiwan

16 Computation Theory Lab., Dept. CSIE, CCU, Taiwan
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17 Thank you.

18 Computation Theory Lab., Dept. CSIE, CCU, Taiwan
References [L94] H. J. Larson, Introduction to Probability, Addison-Wesley Advanced Series in Statistics, 1994; 機率學的世界, 鄭惟厚譯, 天下文化出版. [MR95] R. Motwani and P. Raghavan, Randomized Algorithms, Cambridge University Press, 1995. [H01] 黃文典教授, 機率導論講義, 成大數學系, 2001. Computation Theory Lab., Dept. CSIE, CCU, Taiwan