A Survey of Probability Concepts Chapter 5 Copyright

Learning Objectives LO 5 -1 定義機率 probability, 實驗experiment, 事件event, and 結果outcome. n LO 5 -2 Assign probabilities using a 古典classical, 實證 empirical, or 主觀subjective approach. n LO 5 -3 用加法法則 rules of addition來計算機率 n LO 5 -4 用乘法法則rules of Multiplication 來計算機率 n LO 5 -5 用條件次數/交叉/列聯表contingency table來計算機率 n LO 5 -6用貝氏定理Bayes’ theorem來計算機率 n LO 5 -7 Determine the number of outcomes using principles of counting. n 5 -*

Talus, knucklebones

The oldest known dice were excavated as part of a 5000 year-old backgammon set at the Shahr-e Sukhteh (Burnt City) , an archeological site in south-eastern Iran

LO 5 -1 Define the terms probability, experiment, event, and outcome. 機率Probability PROBABILITY A value between zero and one, inclusive, describing the relative possibility (chance or likelihood) an event will occur. i. e. 0≦P(E) ≦ 1 5 -*

LO 5 -1 Experiment, Outcome, and Event An 實驗(experiment) is a process that leads to the occurrence of one and only one of several possible results. 隨機實驗乃一種過程，是不能確定預知會發生何種結果的實驗方式。 n An 結果/出象(outcome) is the particular result of an experiment. 隨機實驗的每個可能結果都是elementary outcome，又稱為樣本點 (sample point)。一個隨機實驗中，所有可能outcomes的集合稱為：樣本空間(sample space)。 n An 事件(event) is the collection of one or more outcomes of an experiment. 樣本空間的部分集合稱為事件。 n 事件中僅包含 1個outcome或樣本點者，稱為簡單事件(simple event)。 n 事件中包含 2個outcomes或 2個樣本點以上者，稱為複合事件 (composite event)。 n 5 -*

LO 5 -2 Assign probabilities using a classical, empirical, or subjective approach. Ways to Assign Probabilities 我們根據什麼來求得事件發生的機率？ 3種可能方法: 客觀： 1. 古典機率論 CLASSICAL PROBABILITY Based on the assumption that the outcomes of an experiment are equally likely. 如：擲公正的銅板得正面的機率 2. 實證機率論 EMPIRICAL PROBABILITY The probability of an event happening is the fraction of the time similar events happened in the past. 主觀： 3. 主觀機率論 SUBJECTIVE PROBABILITY The likelihood (probability) of a particular event happening that is assigned by an individual based on whatever information is available. 5 -*

LO 5 -2 古典機率論 Classical Probability 不需要做實驗，我們就應該知道擲一個公正的骰子，出現每一面的機率都應該相等，那麼若問：「雙數點出現的機率＝？」 The possible outcomes are: There are three “favorable” outcomes (a two, a four, and a six) in the collection of six equally likely possible outcomes. 5 -*

LO 5 -2 實證機率論 Empirical Probability Empirical approach to probability is based on what is called the Law of Large Numbers. 大數法則 The key to establishing probabilities empirically: a larger number of observations provides a more accurate estimate of the probability. 觀察值（次數）越多（亦即：實驗次數越多），機率的估算越準確。見本次ppt的p. 12 5 -*

LO 5 -2 Empirical Probability - Example On February 1, 2003, the Space Shuttle Columbia exploded. This was the second disaster in 123 space missions for NASA. On the basis of this information, what is the probability that a future mission is successfully completed? 5 -*

LO 5 -2 主觀機率論Subjective Probability - Example n If there is little or no data or information to calculate a probability, it may be arrived at subjectively. Illustrations of subjective probability are: 1. Estimating the likelihood the New England Patriots will play in the Super Bowl next year. 2. Estimating the likelihood a person will be married before the age of 30. 3. Estimating the likelihood the U. S. budget deficit will be reduced by half in the next 10 years. n 5 -*

LO 5 -2 Summarizing Probability 5 -*

LO 5 -3 Calculate probabilities using rules of addition. 加法定理 Rules of Addition for Computing Probabilities §Special Rule of Addition - If two events A and B are mutually exclusive, the probability of one or the other event occurring equals the sum of their probabilities. 互斥 P(A or B) = P(A) + P(B) §Events are mutually exclusive if the occurrence of any one event means that none of the others can occur at the same time. 5 -*

LO 5 -3 Special Rule of Addition- Example Mutually Exclusive Events A machine fills 蔬菜包plastic bags with a mixture of beans, broccoli, and other vegetables. Most of the bags contain the correct weight, but because of the variation in the size of the beans and other vegetables, a package might be underweight or overweight. A check of 4, 000 packages filled in the past month revealed: What is the probability that a particular package will be either underweight or overweight? P(A or C) = P(A) + P(C) =. 025 +. 075 =. 10 5 -*

LO 5 -1 Basic Probability: The Complement Rule 餘集（餘事件）定理 The complement rule is used to determine the probability of an event occurring by subtracting the probability of the event not occurring from 1. P(A) + P(~A) = 1 or P(A) = 1 - P(~A). 5 -*

LO 5 -1 The Complement Rule - Example An experiment has two 互斥 mutually exclusive outcomes. Based on the rules of probability, the sum of the probabilities must be one. If the probability of the first outcome is. 61, then logically, AND by the complement rule, the probability of the other outcome is (1. 0 -. 61) =. 39. P(B) = 1 - P(~B) = 1 –. 61 =. 39 5 -*

LO 5 -3 Complement Rule- Example 互斥 Mutually Exclusive Events The complement rule can also be used: Note that P(A or C) = P(~B), so P(~B) = 1 – P(B) = 1 -. 900 =. 10 5 -*

LO 5 -3 Rules of Addition for Computing Probabilities The General Rule of Addition - If A and B are two events that are 非互斥 not mutually exclusive, then P(A or B) is given by the following formula: P(A or B) = P(A) + P(B) - P(A and B) P( A and B) is called a 聯合機率 joint probability. 5 -*

LO 5 -3 The General Rule of Addition The Venn Diagram shows the results of a survey of 200 tourists who visited Florida during the year. The results revealed that 120 went to Disney World, 100 went to Busch Gardens, and 60 visited both. What is the probability a selected person visited either Disney World or Busch Gardens? (求聯合機率) P(Disney or Busch) = P(Disney) + P(Busch) - P(both Disney and Busch) = 120/200 + 100/200 – 60/200 =. 60 +. 50 –. 80 5 -*

LO 5 -3 General Rule of Addition– Example What is the probability that a card chosen at random from a standard deck of cards will be either a king（老K） or a heart（紅心）? 聯集 P(A or B) = P(A) + P(B) - P(A and B) = 4/52 + 13/52 - 1/52 = 16/52, or. 3077 5 -*

LO 5 -4 Calculate probabilities using the rules of multiplication. Special Rule of Multiplication n The special rule of multiplication calculates the joint probability of two events A and B that are 獨立 independent. Two events A and B are independent if the occurrence of one has no effect on the probability of the occurrence of the other. This rule is written: P(A and B) = P(A)P(B) 5 -*

LO 5 -4 Special Rule of Multiplication. Example A survey by the American Automobile Association (AAA) revealed 60 percent of its members made airline reservations last year. Two members are selected at random. Since the number of AAA members is very large, we can assume that R 1 and R 2 are independent. What is the probability both made airline reservations last year? Solution: The probability the first member made an airline reservation last year is. 60, written as P(R 1) =. 60 The probability that the second member selected made a reservation is also. 60, so P(R 2) =. 60. P(R 1 and R 2) = P(R 1)P(R 2) = (. 60) =. 36 5 -*

LO 5 -4 General Rule of Multiplication The general rule of multiplication is used to find the joint probability that two events will occur when they are 互不獨立not independent. It states that for two events, A and B, the joint probability that both events will happen is found by multiplying the probability that event A will happen by the 條件機率 conditional probability of event B occurring given that A has occurred. 5 -*

LO 5 -4 條件機率 Conditional Probability n A conditional probability is the probability of a particular event occurring, given that another event has occurred. n The probability of the event A given that the event B has occurred is written P(A|B). ：表示事件B發生後，再發生事件A的機率。 5 -*

LO 5 -4 General Rule of Multiplication – Example (p. 148) A golfer has 12 golf shirts in his closet. Suppose 9 of these shirts are white and the others blue. He gets dressed in the dark, so he just grabs a shirt and puts it on. He plays golf two days in a row and does not do laundry. ∵不洗衣服 ∴穿過的衣服的不會再選 What is the likelihood both shirts selected are white? 5 -*

LO 5 -4 General Rule of Multiplication Example n n The probability that the first shirt selected is white is P(W 1) = 9/12. The probability of selecting a second white shirt (W 2 ) is dependent on the first selection. So, the conditional probability is the probability the second shirt selected is white, given that the first shirt selected is also white: P(W 2 | W 1) = 8/11. 第二件衣服只能從剩下的乾淨衣服裡選 Apply the General Multiplication Rule[5 -6]: P(A and B) = P(A) P(B|A) The joint probability of selecting 2 white shirts is: P(W 1 and W 2) = P(W 1)P(W 2 |W 1) = (9/12)(8/11) = 0. 55 5 -*

列聯表 Contingency Tables p. 149

LO 5 -5 Calculate probabilities using a contingency table. 列聯表 Contingency Tables A contingency table is used to classify sample observations according to two or more identifiable characteristics measured. For example, 150 adults are surveyed about their attendance of movies during the last 12 months. Each respondent is classified according to two criteria—the number of movies attended and gender. 5 -*

LO 5 -5 Contingency Table - Example Based on the survey, what is the probability that a person attended zero movies? Based on the empirical information, P(zero movies) = 60/150 = 0. 4 5 -*

LO 5 -5 Contingency Table - Example Based on the survey, what is the probability that a person attended zero movies or is male? (聯集，互斥否？) Applying the General Rule of Addition: P( zero movies or male) = P( zero movies) + P(male) – P(zero movies and male) P(zero movies or male) = 60/150 + 70/150 – 20/150 = 0. 733 5 -*

LO 5 -5 Contingency Table - Example Based on the survey, what is the probability that a person attended zero movies if a person is male? (條件) Applying the concept of conditional probability: P( zero movies | male) = 20/70 = 0. 286 5 -*

LO 5 -5 Contingency Table - Example Based on the survey, what is the probability that a person is male and attended zero movies? (求聯合機率，獨立否？) Applying the General Rule of Multiplication P( male and zero movies) = P(male)P(zero movies|male) = (70/150)(20/70) = 0. 133 5 -*

LO 5 -5 樹形圖 Tree Diagrams A tree diagram is: 1. 用來描繪條件機率、聯合機率非常好用。 Useful for portraying conditional and joint probabilities. 2. 最常被用來分析多階段的企業決策。 Particularly useful for analyzing business decisions involving several stages. 3. 多階段決策中用來計算機率非常好用。 A graph that is helpful in organizing calculations that involve several stages. 每一層討論一個階段的問題，樹枝上的機率為該可能結果的權數。 Each segment in the tree is one stage of the problem. The branches of a tree diagram are weighted by probabilities. 5 -*

p. 153

LO 5 -5 Tree Diagram- Example 抽取 200名經理來調查他們對公司的忠誠度，問卷問題：「若其他公司給你相等或更好一點的職缺，你會留在原來的公司，還是接受其他公司的聘僱？」，問卷結果列於下表：（包含他們年資、與他們的回答） A sample of executives were surveyed about their loyalty to their company. One of the questions was, “If you were given an offer by another company equal to or slightly better than your present position, would you remain with the company or take the other position? ” The responses of the 200 executives in the survey were cross-classified with their length of service with the company. 5 -*

LO 5 -5 Tree Diagram - Example 5 -*

LO 5 -6 Calculate probabilities using Bayes’ theorem. 貝氏定理 Bayes’ Theorem n 若有新的情資，貝氏定理提供一個修正(更新)機率的方法 Bayes’ theorem is a method for revising a probability given additional information. n It is computed using the following formula: 5 -*

前提: 事件Ai之間是互斥及耗盡的關係。事件B是新資訊(additional information) P(Ai) 事前機率(prior probability) 根據現有資訊所得的最初機率 (the initial probability based on the present level of information) P(BIAi) 條件機率事件B的條件機率 P(Ai. IB) 事後機率(posterior probability) 根據新資訊來修正事前機率之後所得的機率 (a revised probability based on additional information)

貝氏定理 Bayes’ Theorem n 貝氏定理乃描述一有先後發生順序的是件，如同樹形機率圖一般，當有新資訊時，貝氏定理便可用來更新先前事件發生的機率。更新後的機率為事後機率。 Bayes' theorem can be understood better by visualizing the events as sequential as depicted in the probability tree. When additional information is obtained about a subsequent event; it is used to revise the probability of the initial event. The revised probability is called posterior. n 換句話說，我們原先有的因果關係模型：可用來預測B發生的可能性（在Ai已經發生的情況下） In other words, we initially have a cause-effect model where we want to predict whether event B will occur or not, given that event Ai has occurred. n 當我們被告知B已經發生，這時，我們便進入推論模式，而我們的目標是Ai究竟是否發生，其機率為何？ We then move to the inference model where we are told that event B has occurred and our goal is to infer whether event Ai has occurred or not n n 總之，貝氏定理提供了一種簡單的方法，可將每個可能肇因（A）的不同影響（結果：B）的機率（ i. e. P(B|Ai) ），轉換成在該影響（結果：B）下，其肇因(A) 可能發生的機率（i. e. P(Ai|B) ）。 In summary, Bayes' Theorem provides us a simple technique to turn information about the probability of different effects (outcomes) from each possible cause, into information about the probable cause given the effect (outcome).

LO 5 -6 Bayes’ Theorem – Example p. 157 5 -*

Bayes’ Theorem – Example p. 157 事前機率 5 -* LO 5 -6

LO 5 -6 Bayes’ Theorem – Example p. 157 條件機率 5 -*

Bayes’ Theorem – Example p. 157 條件機率 joint probability P(A 1∩B) = P(A 1) P(B|A 1)=0. 3 X 0. 03=0. 009 P(A 2∩B) = P(A 2) P(B|A 2)=0. 2 X 0. 05=0. 01 P(A 3∩B) = P(A 3) P(B|A 3)=0. 5 X 0. 04=0. 02 ΣP(Ai ∩B) =0. 039

LO 5 -6 Bayes’ Theorem – Example 事後機率 5 -*

Bayes Theorem – Example (P. 159)

計數法則：p. 160~ 如果可能的outcomes數目很大，有三種方法可以計算outcomes的數目： (1)Multiplication formula乘數法則 (2)Permutation formula 排列法則 (3)Combination formula 組合法則 n

LO 5 -7 Determine the number of outcomes using principles of counting. Counting Rules – Multiplication 乘數法則 The multiplication formula indicates that if there are m ways of doing one thing and n ways of doing another thing, there are m x n ways of doing both. Example: Dr. Delong has 10 shirts and 8 ties. How many shirt and tie outfits does he have? (10)(8) = 80 Official：一實驗包含k次試驗(E 1, E 2, …Ek)，若每一試驗Ei有ni種結果(i=1, 2, 3…k)，則該隨機實驗有 n 1 xn 2 xn 3…. xnk種可能結果 5 -*

LO 5 -7 Counting Rules – Multiplication Example An automobile dealer wants to advertise that for $29, 999 you can buy a convertible, a two-door sedan, or a fourdoor model with your choice of either wire wheel covers or solid wheel covers. How many different arrangements of models and wheel covers can the dealer offer? 5 -*

LO 5 -7 Counting Rules – Permutation 排列法則 A permutation is any arrangement of r objects selected from n possible objects. The order of arrangement is important in permutations. 排列順序很重要 5 -*

排列法則 Permutation formula(p. 169) 排列法則：從一組含有n個元素的集合中，一次抽取r個元素(或每抽取一個，抽出不放回，連續抽r個)，則共有n. Pr 種不同排列。元素排列的順序很重要。 A permutation is any arrangement of r objects selected from n possible objects. The order of arrangement is important in permutations. n! n factorial = n(n-1)x(n-2)x(n-3)x…. (1) 0!=1

LO 5 -7 Counting Rules – Combination 組合法則 A combination is the number of ways to choose r objects from a group of n objects without regard to order. 5 -*

LO 5 -7 Combination and Permutation Examples COMBINATION EXAMPLE There are 12 players on the Carolina Forest High School basketball team. Coach Thompson must pick 5 players among the 12 on the team to comprise the starting lineup. How many different groups are possible? PERMUTATION EXAMPLE Suppose that in addition to selecting the group, he must also rank each of the players in that starting lineup according to their ability. How many different rankings are possible for five players selected from the 12? 5 -*