Business Statistics: A First Course 5th Edition Chapter 4 Basic Probability Business Statistics: A First Course, 5e © 2009 Prentice-Hall, Inc. Chap 4-1 Learning Objectives In this chapter, you learn: Basic probability concepts(概率的基本概念) Conditional probability (条件概率) To use Bayes’ Theorem to revise probabilities (利用贝叶斯定理来修正概率) Business Statistics: A First Course, 5e © 2009 Prentice-Hall, Inc. Chap 4-2 Basic Probability Concepts Probability – the chance that an uncertain event will occur (always between 0 and 1)(一个不确 定事件发生的可能性称为概率) Impossible Event – an event that has no chance of occurring (probability = 0)(如果一个 事件发生概率为零,则称为不可能事件) Certain Event – an event that is sure to occur (probability = 1)(相反,如果事件发生概率为1, 则为必然事件) Business Statistics: A First Course, 5e © 2009 Prentice-Hall, Inc. Chap 4-3 Assessing Probability There are three approaches to assessing the probability of an uncertain event(三种评价事件发生概率的方法): 1. a priori -- based on prior knowledge of the process(先验概率,基于预先掌 握的知识) Assuming all outcomes are equally likely要假定 各种结果出 现的可能性 是相同的 probability of occurrence X number of ways the event can occur T total number of elementary outcomes 2. empirical probability(经验概率,基于观察到的数据) probability of occurrence number of ways the event can occur total number of elementary outcomes 3. subjective probability(主观概率,每个人的主观可能不一样) based on a combination of an individual’s past experience, personal opinion, and analysis of a particular situation 依赖于每 个人不同的经验、观点和对特定情形分析的结合 Example of a priori probability Find the probability of selecting a face card (Jack, Queen, or King) from a standard deck of 52 cards. (52张牌中一次抽到人牌的概率) X number of face cards Probabilit y of Face Card T total number of cards X 12 face cards 3 T 52 total cards 13 Business Statistics: A First Course, 5e © 2009 Prentice-Hall, Inc. Chap 4-5 Example of empirical probability Find the probability of selecting a male taking statistics from the population described in the following table(依照下表描 述的总体,计算一次抽到选择统计的男生的概率): Taking Stats Not Taking Stats Total Male 84 145 229 Female 76 134 210 160 279 439 Total Probability of male taking stats Business Statistics: A First Course, 5e © 2009 Prentice-Hall, Inc. numberof males taking stats 84 0.191 totalnumberof people 439 Chap 4-6 Events Each possible outcome of a variable is an event.(变量的每一 个可能结果称为事件) Simple event(简单事件) Joint event(联合事件) An event described by a single characteristic(对应单一特征) e.g., A red card from a deck of cards An event described by two or more characteristics(两个或者两个以 上特征的时间) e.g., An ace that is also red from a deck of cards Complement of an event A (denoted A’)(事件A的补) All events that are not part of event A(所有不包含事件A的事件集合) e.g., All cards that are not diamonds Business Statistics: A First Course, 5e © 2009 Prentice-Hall, Inc. Chap 4-7 Sample Space(样本空间) The Sample Space is the collection of all possible events(所有可能事件的集合称为样本空间) e.g. All 6 faces of a die: e.g. All 52 cards of a bridge deck: Business Statistics: A First Course, 5e © 2009 Prentice-Hall, Inc. Chap 4-8 Visualizing Events Contingency Tables(列联表) Ace Sample Space Not Ace Total Black 2 24 26 Red 2 24 26 Total 4 48 52 Decision Trees(决策树) 2 Sample Space 24 Full Deck of 52 Cards Business Statistics: A First Course, 5e © 2009 Prentice-Hall, Inc. 2 24 Chap 4-9 Visualizing Events Venn Diagrams(韦恩图示法) Let A = aces Let B = red cards A ∩ B = ace and red A A U B = ace or red Business Statistics: A First Course, 5e © 2009 Prentice-Hall, Inc. B Chap 4-10 Definitions Simple vs. Joint Probability Simple Probability refers to the probability of a simple event.(简单事件发生的概率称为简单概率) ex. P(King) ex. P(Spade黑桃) Joint Probability refers to the probability of an occurrence of two or more events (joint event).(联合事件发生的概率 称为联合概率) ex. P(King and Spade) Business Statistics: A First Course, 5e © 2009 Prentice-Hall, Inc. Chap 4-11 Mutually Exclusive Events(互斥事件) Mutually exclusive events Events that cannot occur simultaneously(不能同时发生的事 件) Example: Drawing one card from a deck of cards A = queen of diamonds方块; B = queen of clubs梅花 Events A and B are mutually exclusive Business Statistics: A First Course, 5e © 2009 Prentice-Hall, Inc. Chap 4-12 Collectively Exhaustive Events(完备事件集) Collectively exhaustive events One of the events must occur(事件集中某个事件一定发生) The set of events covers the entire sample space(整个事 件集覆盖整个样本空间) example: A = aces; B = black cards; C = diamonds; D = hearts Events A, B, C and D are collectively exhaustive (but not mutually exclusive – an ace may also be a heart)(完备事 件集中事件不一定互斥) Events B, C and D are collectively exhaustive and also mutually exclusive Business Statistics: A First Course, 5e © 2009 Prentice-Hall, Inc. Chap 4-13 Computing Joint and Marginal Probabilities 联合概率及边际概率的计算 The probability of a joint event, A and B: number of outcomes satisfyingA and B P( A and B) total number of elementary outcomes Computing a marginal (or simple) probability: P(A) P(A and B1) P(A and B2 ) P(A and Bk ) Where B1, B2, …, Bk are k mutually exclusive and collectively exhaustive events(互斥且完备事件集) Business Statistics: A First Course, 5e © 2009 Prentice-Hall, Inc. Chap 4-14 Joint Probability Example P(Red and Ace) number of cards that are red and ace 2 total number of cards 52 Type Color Red Black Total Ace 2 2 4 Non-Ace 24 24 48 Total 26 26 52 Business Statistics: A First Course, 5e © 2009 Prentice-Hall, Inc. Chap 4-15 Marginal Probability Example P(Ace) P( Ace and Re d) P( Ace and Black ) Type 2 2 4 52 52 52 Color Red Black Total Ace 2 2 4 Non-Ace 24 24 48 Total 26 26 52 Business Statistics: A First Course, 5e © 2009 Prentice-Hall, Inc. Chap 4-16 Marginal & Joint Probabilities In A Contingency Table Event B1 Event B2 Total A1 P(A1 and B1) P(A1 and B2) A2 P(A2 and B1) P(A2 and B2) P(A2) Total P(B1) Joint Probabilities Business Statistics: A First Course, 5e © 2009 Prentice-Hall, Inc. P(B2) P(A1) 1 Marginal (Simple) Probabilities Chap 4-17 Probability Summary So Far Probability is the numerical measure of the likelihood that an event will occur(概率是刻 画一个事件发生可能性的数字测度) Certain The probability of any event must be between 0 and 1, inclusively(任何事件发生 概率值都在0和1之间) 0 ≤ P(A) ≤ 1 For any event A 1 0.5 The sum of the probabilities of all mutually exclusive and collectively exhaustive events is 1(互斥且完备事件集的概率之和为1) P(A) P(B) P(C) 1 If A, B, and C are mutually exclusive and collectively exhaustive Business Statistics: A First Course, 5e © 2009 Prentice-Hall, Inc. 0 Impossible Chap 4-18 General Addition Rule(一般加法原则) General Addition Rule: P(A or B) = P(A) + P(B) - P(A and B) If A and B are mutually exclusive, then P(A and B) = 0, so the rule can be simplified: P(A or B) = P(A) + P(B) For mutually exclusive events A and B Business Statistics: A First Course, 5e © 2009 Prentice-Hall, Inc. Chap 4-19 General Addition Rule Example P(Red or Ace) = P(Red) +P(Ace) - P(Red and Ace) = 26/52 + 4/52 - 2/52 = 28/52 Type Color Red Black Total Ace 2 2 4 Non-Ace 24 24 48 Total 26 26 52 Business Statistics: A First Course, 5e © 2009 Prentice-Hall, Inc. Don’t count the two red aces twice! Chap 4-20 Computing Conditional Probabilities 条件概率的计算 A conditional probability is the probability of one event, given that another event has occurred:一个事件在给定另一事件发生下发生的 概率 P(A and B) P(A | B) P(B) The conditional probability of A given that B has occurred P(A and B) P(B | A) P(A) The conditional probability of B given that A has occurred Where P(A and B) = joint probability of A and B(A、B的联合概率) P(A) = marginal or simple probability of A(A的边际概率) P(B) = marginal or simple probability of B(B的边际概率) Business Statistics: A First Course, 5e © 2009 Prentice-Hall, Inc. Chap 4-21 Conditional Probability Example Of the cars on a used car lot, 70% have air conditioning (AC) and 40% have a CD player (CD). 20% of the cars have both.在某个二手车市场,70%的车有空调,40%的 车有CD机,20%的车这两样东西都有 What is the probability that a car has a CD player, given that it has AC ?(问在那些有空调的车中,拥有CD机的 概率?) i.e., we want to find P(CD | AC) Business Statistics: A First Course, 5e © 2009 Prentice-Hall, Inc. Chap 4-22 Conditional Probability Example (continued) Of the cars on a used car lot, 70% have air conditioning (AC) and 40% have a CD player (CD). 20% of the cars have both. CD No CD Total AC 0.2 0.5 0.7 No AC 0.2 0.1 0.3 Total 0.4 0.6 1.0 P(CD and AC) 0.2 P(CD | AC) 0.2857 P(AC) 0.7 Business Statistics: A First Course, 5e © 2009 Prentice-Hall, Inc. Chap 4-23 Conditional Probability Example (continued) Given AC, we only consider the top row (70% of the cars). Of these, 20% have a CD player. 20% of 70% is about 28.57%. CD No CD Total AC 0.2 0.5 0.7 No AC 0.2 0.1 0.3 Total 0.4 0.6 1.0 P(CD and AC) 0.2 P(CD | AC) 0.2857 P(AC) 0.7 Business Statistics: A First Course, 5e © 2009 Prentice-Hall, Inc. Chap 4-24 Using Decision Trees .2 .7 Given AC or no AC: .5 .7 All Cars P(AC and CD) = 0.2 P(AC and CD’) = 0.5 Conditional Probabilities .2 .3 .1 .3 Business Statistics: A First Course, 5e © 2009 Prentice-Hall, Inc. P(AC’ and CD) = 0.2 P(AC’ and CD’) = 0.1 Chap 4-25 Using Decision Trees .2 .4 Given CD or no CD: .2 .4 All Cars (continued) P(CD and AC) = 0.2 P(CD and AC’) = 0.2 Conditional Probabilities .5 .6 .1 .6 Business Statistics: A First Course, 5e © 2009 Prentice-Hall, Inc. P(CD’ and AC) = 0.5 P(CD’ and AC’) = 0.1 Chap 4-26 Independence(独立性) Two events are independent if and only if: P(A | B) P(A) Events A and B are independent when the probability of one event is not affected by the fact that the other event has occurred(两个事件A、B是相互独立的, 如果A是否发生不受B已经发生的影响) Business Statistics: A First Course, 5e © 2009 Prentice-Hall, Inc. Chap 4-27 Multiplication Rules(乘法法则) Multiplication rule for two events A and B: P(A and B) P(A | B) P(B) Note: If A and B are independent, then P(A | B) P(A) and the multiplication rule simplifies to P(A and B) P(A)P(B) Business Statistics: A First Course, 5e © 2009 Prentice-Hall, Inc. Chap 4-28 Marginal Probability Marginal probability for event A: P(A) P(A | B1)P(B1) P(A | B2 )P(B 2 ) P(A | Bk )P(Bk ) Where B1, B2, …, Bk are k mutually exclusive and collectively exhaustive events 化边际概率的计算为多个条件概率乘积的和 Business Statistics: A First Course, 5e © 2009 Prentice-Hall, Inc. Chap 4-29 Bayes’ Theorem Bayes’ Theorem is used to revise previously calculated probabilities based on new information.(基于新的信息修正原先已经计算 的概率) Developed by Thomas Bayes in the 18th Century.(由托马斯-贝叶斯发展于18世纪) It is an extension of conditional probability. (条件概率的一个推广) Business Statistics: A First Course, 5e © 2009 Prentice-Hall, Inc. Chap 4-30 Bayes’ Theorem(贝叶斯定理) P(A| B i )P(Bi ) P(Bi | A) P(A| B 1 )P(B1 ) P(A| B 2 )P(B2 ) P(A| B k )P(Bk ) where: Bi = ith event of k mutually exclusive and collectively exhaustive events( Bi ,k个互斥且完备的事件) A = new event that might impact P(Bi) Business Statistics: A First Course, 5e © 2009 Prentice-Hall, Inc. Chap 4-31 Bayes’ Theorem Example A drilling company has estimated a 40% chance of striking oil for their new well. 一家石油钻探公司新钻井出油成功率大概在40%,可以看成 先验成功率 A detailed test has been scheduled for more information. Historically, 60% of successful wells have had detailed tests, and 20% of unsuccessful wells have had detailed tests. 详细的钻前勘探能带来更 多信息。历史上成功出油的钻井中60%做过详细的钻前勘探,没有出油 的钻井中20%做过详细的钻前勘探 Given that this well has been scheduled for a detailed test, what is the probability that the well will be successful?(假定该次钻探前,已经做 了详细的钻前勘探,请问该井成功出油的概率?) Business Statistics: A First Course, 5e © 2009 Prentice-Hall, Inc. Chap 4-32 Bayes’ Theorem Example (continued) Let S = successful well U = unsuccessful well P(S) = 0.4 , P(U) = 0.6 Define the detailed test event as D Conditional probabilities: P(D|S) = 0.6 (prior probabilities) P(D|U) = 0.2 Goal is to find P(S|D) Business Statistics: A First Course, 5e © 2009 Prentice-Hall, Inc. Chap 4-33 Bayes’ Theorem Example (continued) Apply Bayes’ Theorem: P(D | S)P(S) P(S | D) P(D | S)P(S) P(D | U)P(U) (0.6)(0.4) (0.6)(0.4) (0.2)(0.6) 0.24 0.667 0.24 0.12 So the revised probability of success, given that this well has been scheduled for a detailed test, is 0.667 Business Statistics: A First Course, 5e © 2009 Prentice-Hall, Inc. Chap 4-34 Bayes’ Theorem Example (continued) Given the detailed test, the revised probability of a successful well has risen to 0.667 from the original estimate of 0.4 Event Prior Prob. Conditional Prob. Joint Prob. Revised Prob. S (successful) 0.4 0.6 (0.4)(0.6) = 0.24 0.24/0.36 = 0.667 U (unsuccessful) 0.6 0.2 (0.6)(0.2) = 0.12 0.12/0.36 = 0.333 Sum = 0.36 Business Statistics: A First Course, 5e © 2009 Prentice-Hall, Inc. Chap 4-35 Chapter Summary Discussed basic probability concepts Examined basic probability rules General addition rule, addition rule for mutually exclusive events, rule for collectively exhaustive events Defined conditional probability Sample spaces and events, contingency tables, Venn diagrams, simple probability, and joint probability Statistical independence, marginal probability, decision trees, and the multiplication rule Discussed Bayes’ theorem Business Statistics: A First Course, 5e © 2009 Prentice-Hall, Inc. Chap 4-36