Statistics Assignment #3 Due: 10/16/2014 Perform the following questions manually (請寫公式，不可直接寫答案) 學習重點: 1. 相對機率 (Relative Frequency Approach) 2. 條件機率 (Conditional Probability) 3. 機率樹 (Probability Tree) 4. 貝氏定理 (Bayes’ Law) 1. A Morgan Stanley Consumer Research Survey sampled men and women and asked each whether the preferred to drink plain bottled water or a sports drink such as Gatorade or Propel Fitness water (The Atlanta Journal –Constitution, December 28, 2005). Suppose 200 men and 200 women participated in the study, and 280 reported they preferred plain bottled water. Of the group preferring a sports drink, 80 were men and 40 were women. Let M=the event the consumer is a man W= the event the consumer is a woman B=the event the consumer Preferred plain bottled water S= the event the consumer Preferred sports drink. a. b. c. d. e. f. g. Ans: What is the probability a person in the study preferred plain bottled water? What is the probability a person in the study preferred a sports drink? What are the conditional probabilities P(M|S) and P(W|S)? What are the joint probabilities P(M  S) and P(W  S)? Give a consumer is a man, what is the probability he will prefer a sports drink? Give a consumer is a woman, what is the probability she will prefer a sports drink? Is preference for a sports drink independent of whether the consumer is a man or woman? Explain using probability information. 2. The prior probabilities for events A1 and A2 are P( A1 )=.40 and P( A2 )=.60. It is also know that P( A1  A2 )=0. Suppose P(B| A1 )=.20 and P(B| A2 )=.05. a. Are A1 and A2 mutually exclusive? Explain. b. Compute P( A1  B) and P( A2  B). c. Compute P(B) d. Apply Bayes’ theorem to compute P( A1 |B) and P( A2 |B) Ans: a. Yes, since P( A1  A2 )  0 3. To help select suitable employees for a particular job a personnel department administers and aptitude test to all applicants. To test the effectiveness of the test a sample of applicants who failed were also hired and given a fast trial at the job. It was found that of the 30 percent who passed the test, 80% were satisfactory, and of those who did not, only 10 percent were satisfactory a. What is the probability that an applicant selected at random will prove to be satisfactory at this job? b. If an applicant is satisfactory, what is the probability that he passed the test? Ans: Assume T is the event of passing the test, and E is the event of satisfactory Then a. P(E)= P(T ) P( E T )  P(T c ) P( E T c ) = 0.3 0.8  0.7  0.1  0.31 b. P(T E )  0.3  0.8  0.7742 0.3  0.8  0.7  0.1 4. Your favorite team is in the final playoffs. You have assigned a probability of 60% that they will win the championship. Past records indicate that when team win the championship, the win the first game of the series 70% of the time. When they lose the series, they win the first game 25% of the time. The first game is over; your team has lost. what is the probability that they win the series? Ans: 5. 最近網路上很流行”理想情人 vs 恐怖情人” app，其主要透過貝氏定理來判定，目前喜歡的對象是否為適合的伴侶。假設最近有朋友要介紹小美給小明認識，小明在尚未見過小美前透過朋友描述下，小明認為小美有 35%機率是理想情人。假設 app 告知下列訊息。假設女孩為理想情人，且此女孩第一次約會選擇百貨公司地下街吃飯的機率為 90%，假設女孩不是理想情人，且此女孩第一次約會選擇百貨公司地下街吃飯，機率為 70%。 a. 第一次約會時，小美選擇遠東百貨地下街吃飯，則小美是理想情人的機率為多少? Ans: A:小美是理想情人, B=在地下街吃飯事前機率: 小美是理想情人 P(A)=35% , P(Ac)=65% 給定 P(B|A)=90%, P(B|Ac)=70% (a) 求 P(A|B)= ( | ) ( | ) ( ) ( ) ( | ) ( ) 6. Three airlines serve a small town in Indiana. Airline A has 60% of all the scheduled flights, airline B has 30%, and airline C has the remaining 10%. Their on-time rates are 80%, 60%, and 40% respectively. Define event O as an airline arrives on time. a. Calculate P(A and O). b. Calculate the probability that a plane leaves on time c. If a plane has just left on time, what is the probability that it was airline A? d. If a plane has just left 40 minutes late, what is the probability that it was airline A? ANS: (b) P(A and O) = P(A)P(O|A) = (.60)(.80) = 0.48 (c) P(O) = P(A and O) + P(B and O) + P(C and O) = .48 + .18 + .04 = 0.70 (d) P(A|O) = P(A and O) / P(O) = 0.48 / 0.70 = 0.686 (e) P(A|Oc) = P(A and Oc) / P(Oc) = (0.60)(0.20) / 0.30 = 0.40 7. NTU Company tracks the number of desktop computer systems it sells over a year (360 days), assume they take the two-day order policy, please use the following information to decide the best order number so that they can satisfy 70% customers’ need. Note: for example 26 days out of 360, 2 desktops were sold . Desktops sold # of days 0 2 1 6 2 26 3 50 4 60 5 58 6 54 7 34 8 10 Ans : at least order 5 desktops to match the demand. 兩天 demand 兩天# of days 相對機率累加機率 0 25 0.069444 0.069444 1 35 0.097222 0.166667 2 26 0.072222 0.238889 3 50 0.138889 0.377778 4 63 5 58 6 54 7 39 0.175 0.552778 0.161111 0.713889 0.15 0.863889 0.108333 0.972222 8 10 0.027778 360 1 1