Experiment

1. Experiment 定義 • An experiment is any activity from which an outcome, measurement, or result is obtained. 任何求結果的過程或活動皆可稱為「試驗」。 • When the outcomes cannot be predicted with certainty, the experiment is a random experiment.當結果無法事先預測時，稱此試驗為「隨機試驗」。 社會統計（上）

2. Example of Experiments 實例 • 1. Measuring the lifetime (time to failure) of a given product • 2. Inspecting an item to determine whether it is defective • 3. Recording the income of a bank employee • 4. Recording the balance in an individual's checking account 社會統計（上）

3. Basic Outcomes and Sample Space 定義 • The set of all possible basic outcomes for a given experiment is called the sample space.隨機實驗的所有可能結果稱為樣本空間，一般以S或Ω表示。 • Each possible outcome of a random experiment is called a basic outcome (or a sample point, an element in the sample space). 隨機實驗中所得到的任何可能的個別結果稱之為「基本結果」（或樣本點、或簡稱樣本），以小寫oi表示。 社會統計（上）

4. Basic Outcomes and Sample Space 實例 • 某家公司在六個不同的城市有辦公室，某新進員工將被分派到其中的一個辦公室上班，則所有可能的分派為： • o1= 台北 o2= 桃園 o3= 台中 • o4= 台南 o5= 高雄 o6= 屏東 • 分派的樣本空間為： • S = { o1 , o2 , o3 , o4 ,o5 ,o6 } • 如果這名員工被分派至台南上班，我們說該試驗的結果為 o4 =台南，即該試驗的o4結果發生了。 社會統計（上）

5. Venn Diagrams 定義 • S = { o1 , o2 , o3 , o4 ,o5 ,o6 } ‧ ‧ ‧ o1o2 o3 o4o5o6 ‧ ‧ ‧ 社會統計（上）

6. Event事件 定義 • An event is a specific collection of basic outcomes, that is, a set containing one or more of the basic outcomes from the sample space. We say that an event occurs if any one of the basic outcomes in the event occurs. • 事件為包含樣本空間中一個以上樣本元素（基本結果）的子集合，即樣本空間的任何部分集合(subset)，一般以英文大寫字母表示。 社會統計（上）

7. Event事件 實例 定義 • 上例中，該員工被分派至南台灣的事件為： • A = {o4 ,o5 ,o6 } • 如果該員工被分派至台南上班，我們可以說A事件發生了。 • 若B事件為該員工被分派至直轄市上班，則 • B = {o1 ,o5} 社會統計（上）

8. Venn Diagrams 定義 • S = { o1 , o2 , o3 , o4 ,o5 ,o6 } B事件 ‧ ‧ ‧ o1o2 o3 o4o5o6 ‧ ‧ ‧ A事件 社會統計（上）

9. Event事件 實例 • 寫出丟骰子一次所得數目的樣本空間： • S = {1, 2, 3, 4, 5, 6} • 寫出得到偶數的事件： • A = {2,4,6} • 寫出數字大於2的事件： • B = {3,4,5,6} 社會統計（上）

10. Event事件 實例 • 丟銅板得到H正面或T反面，寫出丟一銅板三次的樣本空間： • S = {HHH, HHT, HTH, HTT, THH, THT, TTH, TTT} • 設A表示第一次出現正面的事件，則 • A = {HHH, HHT, HTH, HTT} 社會統計（上）

11. Assigning Probabilities to Events事件的機率 定義 • There are two types of random experiments, those that can be repeated over and over again under essentially identical conditions and those that are unique and cannot be repeated. 社會統計（上）

12. Assigning Probabilities to Events事件的機率 定義 • A numerical measure that indicates the likelihood of a specific outcome in a repeatable random experiment is called an objective probability, whereas the probability associated with a specific outcome of a unique and nonrepeatable random experiment is called a subjective probability. 社會統計（上）

13. Assigning Probabilities to Events事件的機率 定義 • There are three different approaches to assigning probabilities to basic outcomes: • 1. The relative frequency approach • 2. The equally likely approach • 3. The subjective approach 社會統計（上）

14. The Relative Frequency Approach 相對次數（後天）機率理論 定義 • Let fAbe the number of occurrences, or frequency of occurrence, of event A in n repeated identical trials. The probability that A occurs is the limit of the ratio fA/n as the number of trials n becomes infinitely large. 社會統計（上）

15. The Relative Frequency Approach 相對次數（後天）機率理論 定義 • 當一試驗在完全相同的情境中重複試驗，某事件A發生的機率可以定義為：當試驗在相同情境下重複無限次時，A所發生的次數與試驗總次數之比。 • 由於這種機率為長期試驗的結果，因此又稱之為後天機率、相對次數。由於這種機率為歸納多次試驗的結果所得，故又稱為統計機率或經驗機率。 社會統計（上）

16. 後天機率理論的缺陷 定義 • (1) Because we can never replicate an experiment an infinite number of times, it is impossible to determine the limit of the ratio fA/n as n approaches infinity. • (2) We can never be sure that we have repeated an experiment under identical conditions. 社會統計（上）

17. 後天機率理論的缺陷 定義 • When we use the relative frequency approach, we use the observed ratio fA/n to approximate the theoretical probability that event A occurs. That is, we assume that P(A) fA/n when n is sufficiently large. 社會統計（上）

18. The Equally Likely Approach古典（先天）機率理論 定義 • Suppose that an experiment must result in one of n equally likelyoutcomes. Then each possible basic outcome is considered to have probability 1/n of occurring on any replication of the experiment. • 一樣本空間有n個樣本點（基本結果），且每一個樣本點發生的機會皆相等。則在任何重複試驗中，每一個樣本點發生的機率為1/n。若事件A的樣本點個數為nA，則A發生的機率為： • P(A) = nA/n，符合某條件結果的個數與總結果數之比。 社會統計（上）

19. The Equally Likely Approach古典（先天）機率理論 實例 • 直一骰子兩次，求點數和等於6的機率？ • E = { (1,5) (2,4) (3,3) (4,2) (5,1)} • P(E) = 5/36 • 其他先天機率：丟銅板、抽籤、彩券等。 • 先天機率最重要的問題：所有樣本點發生的機率是否相同？ 社會統計（上）

20. Objective Probability客觀機率 定義 • A probability obtained by using a relative frequency approach or an equally likely approach is called an objective probability. 社會統計（上）

21. The Subjective Approach主觀機率 定義 • 如果樣本點出現的可能性不等，則無法求事件的先天機率。若試驗無法進行，或無法多次重複，則相對次數無法計算，事件的後天機率也無法計算。 • 在此情況下，我們經常仰賴主觀機率，即人們對於某事件發生的主觀評價，或可以看成是「專家」的意見。 • A subjective probability is a number in the interval [0, 1] that reflects a person's degree of belief that an event will occur. 社會統計（上）

22. Odds可能性 定義 • 有時候人們以「發生」與「不發生」的比值來陳述他們對於事件發生機率的意見。 • 如經常聽到某人說某賭局、競賽、或遊戲有3:1的勝算，則表示某人有75%贏的機率。 • If the odds in favor of event A occurring are a to b, then 社會統計（上）

23. Which approach is best？ 定義 • The nature of the problem determines which approach is best. • Problems with an underlying symmetry, such as coin, dice, and card problems, are especially suited to the equally likely approach. • Problems for which we have large samples of data based on many replications of an experiment are especially suited to the relative frequency approach. • Problems that occur only once, such as a sporting event, are especially suited to the subjective approach. 社會統計（上）

24. Which approach is best？ 實例 • 已知懷單胎，生男的機率為何？ • (1) 依先天機率： • 性別有男女兩類，故生男的機率為1/2 • (2) 依後天機率： • 台灣地區男性人口佔52%，故生男的機率為52% • (3) 主觀機率： • 由於飲食習慣的改變，現代人大量攝取動物性蛋白質，使生男的機率增加為60% 社會統計（上）

25. Set Theory 定義 • Subset子集合 • An event A is contained in another event B if every outcome that belongs to the subset defining the event A also belongs to the subset defining the event B. • A = {2,4,6} B={2,3,4,5,6} • A  B, A is a subset of B • If A  B and B  A, then A = B • If A  B and B  C, then A  C • Empty Set or Null Set空集合Ø • For any event A, Ø  A  S, 社會統計（上）

26. Operation of Set Theory: Unions聯集 定義 • Unions聯集 • Let A and B be two events in the sample space S. Their union, denoted A U B. is the event composed of all basic outcomes in S that belong to at least one of the two events A or B. Hence, the union A U B occurs if either A or B (or both) occurs. S A B 社會統計（上）

27. Operation of Set Theory: Unions聯集 定義 • Unions聯集: The union of n events A1,A2,…,An is defined to be the event that contains all outcomes which belong to at least one of these n events. S A B 社會統計（上）

28. Operation of Set Theory: Intersection交集 定義 • Intersection交集: • Let A and B be two events in the sample space S. The intersection of A and B, denoted A  B. is the event composed of all basic outcomes in S that belong to both A and B. Hence, the intersection A  B occurs if both A and B occur. S A B 社會統計（上）

29. Operation of Set Theory: Intersection交集 定義 • Intersection交集: • The intersection of n events, A1, …An is defined to be the event that contains the outcomes which are common to all these n events. S A B 社會統計（上）

30. Complement of an Event 定義 • Let A denote some event in the sample space S. The complement of A (A的餘事件）, denoted by Ac, represents the event composed of all basic outcomes in S that do not belong to A. S A Ac 社會統計（上）

31. Complement has the following properties: 定義 • (Ac)c =A • A  Ac = S • Øc = S • Sc = Ø • A  Ac = Ø S A Ac 社會統計（上）

32. Complement has the following properties: 定義 S • (A ∪ B)c =Ac ∩Bc • (A ∩ B)c =Ac ∪Bc Bc B A Ac • P(A) = P(A ∩ B) +P(A ∩Bc) • P(Ac ∩ Bc) = 1 － P(A ∪B) • P(Ac ∪ Bc) = 1 － P(A ∩B) 社會統計（上）

33. Mutually Exclusive Events (Disjoint Events) • Let A and B be two events in a sample space S. If A and B have no basic outcomes in common, then they are said to be mutually exclusive. If A and B are mutually exclusive events, we write (A  B) = Ø, where Ø denotes the empty set. P(A  B) = 0. 社會統計（上）

34. Some basic rules of probability 定義 • Probability of a basic outcome: • For each basic outcome oi, 0 P(oi)  1. • Probability of an event: • Let event A= { o1 , o2 , o3 , o4 ,o5 ,…ok }, where o1 , o2 , o3 , o4 ,o5 ,…ok are kdifferent basic outcomes. The probability of any event A is the sum of the probabilities of the basic outcomes in A. That is, • P(A) = P(o1) + P(o2) + P(o3) + P(o4) + P(o5) +…+P(ok) = AP(oi) • where AP(oi) means to obtain the sum over all basic outcomes in event A. 社會統計（上）

35. Some basic rules of probability 定義 • Probability of an Event • For any event A, 0  P(A)  1. • Probability of a sample space: • Let event S = { o1 , o2 , o3 , o4 ,o5 ,…on } represent the sample space of an experiment. The probability of S is • P(S) = sP(oi) = 1 社會統計（上）

36. Definition of Probability • Axiom 1公理1: For any event A, P(A)  0事件A發生的機率為實數 • Axiom 2: P(S) = 1. • Axiom 3: For any infinite sequence of disjoint events (互斥事件） A1, A2, … 社會統計（上）

37. Definition of Probability • A probability distribution , or simply a probability, on a sample space S is a specification of numbers P(A) which satisfy Axioms 1,2, and 3. • 設有一試驗其樣本空間為S，對S中之任一事件A指定一值P(A)，若P(.)滿足上述三個公理，則稱P（.）為一機率測度，且稱P(A) 為事件A的機率。 社會統計（上）

38. Theorem 1:零事件的機率 • 零事件的機率等於零 • Pr(Ø)=0 社會統計（上）

39. Theorem 2: • For any finite sequence of ndisjoint eventsA1,A2,…,An 與公理3的不同處 社會統計（上）

40. Theorem 2: • Proof. 假設一組無窮盡的事件序列A1, A2, A3,…, 其中A1… An為互斥事件, Ai = , i > n 社會統計（上）

41. Theorem 3:餘事件的機率Probability of the complement of an event 定義 • Let Ac denote the complement of A. Then P(Ac) = 1 – P(A). • Proof: • Since A and Ac are disjoint events and A  Ac = S, • it follows from Theorem 2 that P(S) = P(A) + P(Ac). • Since P(S) =1 by Axiom 2, • then P(Ac) = 1 – P(A). 社會統計（上）

42. Theorem 4:機率的範圍 定義 • For any event A, 0≦ P(A) ≦ 1. • Proof. • 從公理1得知 P(A) ≧ 0. • 如果 P(A) > 1 • 則 P(Ac) < 0 →違反公理1 • 所以 P(A) ≦ 1 社會統計（上）

43. Theorem 5 定義 • If A  B, then P(A) ≦ P(B) • Proof. • B = A  BAc • P(B) = P(A)  P(BAc ) • P(BAc ) ≧ 0 • P(A) ≦ P(B) B BAc A 社會統計（上）

44. Theorem 6: • P(A  B) = P(A) + P(B) – P(AB) • Proof: • P(A  B) = P(ABc) + P(AB) + P(AcB) • P(A) = P(ABc) + P(AB) • P(B) = P(AcB)+ P(AB) A B ABC AB ACB 社會統計（上）

45. Theorem 6: • P(A1∪ A2 ∪A3) = P(A1) + P(A2) + P(A3) – [ P(A1∩ A2 ) + P(A2∩ A3 ) + P(A1∩ A3 ) ] + P(A1∩ A2 ∩ A3 ) 社會統計（上）

46. 例題 • Suppose that 15% of the freshmen fail chemistry, • 12% fail math, • and 5% fail both. • Suppose a first‑year student is picked at random. Find the probability that the student failed at least one of the courses. • P(A  B) = P(A) + P(B) – P(AB) • = .15 + .12 － .05 =.22 社會統計（上）

47. Conditional Probability條件機率 定義 • P(AB) : The probability that some event A occurs given that some other event B has already occurred. • If the probability of one event varies depending on whether a second event has occurred, the two events are said to be dependent. 社會統計（上）

48. Conditional Probability條件機率 • A : 大學畢業生 • B：年薪五萬元 • P(B) 年薪五萬元的機率  P(B  A) 大學畢業生年薪五萬的機率 社會統計（上）

49. Joint Probability Tables聯合機率表 row sum column sum 社會統計（上）

50. Joint Probability Tables聯合機率表 男生且被拒絕的機率= 4700/12500 =.376 A joint probability shows the probability that an observation will possess two (or more) characteristics simultaneously. Every joint probability must be a number in the closed interval [0,1] and the sum of all joint probabilities must be 1. 社會統計（上）