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www.downloadslide.com 2.11 Numerical Events and Random Variables 75 selecting the correct answer is .25. If the student correctly answers a question, what is the probability that the student really knew the correct answer? 2.134 Two methods, A and B, are available for teaching a certain industrial skill. The failure rate is 20% for A and 10% for B. However, B is more expensive and hence is used only 30% of the time. (A is used the other 70%.) A worker was taught the skill by one of the methods but failed to learn it correctly. What is the probability that she was taught by method A? 2.135 Of the travelers arriving at a small airport, 60% fly on major airlines, 30% fly on privately owned planes, and the remainder fly on commercially owned planes not belonging to a major airline. Of those traveling on major airlines, 50% are traveling for business reasons, whereas 60% of those arriving on private planes and 90% of those arriving on other commercially owned planes are traveling for business reasons. Suppose that we randomly select one person arriving at this airport. What is the probability that the person a b c d is traveling on business? is traveling for business on a privately owned plane? arrived on a privately owned plane, given that the person is traveling for business reasons? is traveling on business, given that the person is flying on a commercially owned plane? 2.136 A personnel director has two lists of applicants for jobs. List 1 contains the names of five women and two men, whereas list 2 contains the names of two women and six men. A name is randomly selected from list 1 and added to list 2. A name is then randomly selected from the augmented list 2. Given that the name selected is that of a man, what is the probability that a woman’s name was originally selected from list 1? 2.137 Five identical bowls are labeled 1, 2, 3, 4, and 5. Bowl i contains i white and 5 − i black balls, **with** i = 1, 2,..., 5. A bowl is randomly selected and two balls are randomly selected (**with**out replacement) from the contents of the bowl. a b What is the probability that both balls selected are white? Given that both balls selected are white, what is the probability that bowl 3 was selected? *2.138 Following is a description of the game of craps. A player rolls two dice and computes the total of the spots showing. If the player’s first toss is a7oran11,theplayer wins the game. If the first toss is a 2, 3, or 12, the player loses the game. If the player rolls anything else (4, 5, 6, 8, 9 or 10) on the first toss, that value becomes the player’s point. If the player does not win or lose on the first toss, he tosses the dice repeatedly until he obtains either his point or a 7. He wins if he tosses his point before tossing a 7 and loses if he tosses a 7 before his point. What is the probability that the player wins a game of craps? [Hint: Recall Exercise 2.119.] 2.11 Numerical Events and Random Variables Events of major interest to the scientist, engineer, or businessperson are those identified by numbers, called numerical events. The research physician is interested in the event that ten of ten treated patients survive an illness; the businessperson is interested in the event that sales next year will reach $5 million. Let Y denote a variable to be measured in an experiment. Because the value of Y will vary depending on the outcome of the experiment, it is called a random variable. To each point in the sample space we will assign a real number denoting the value of the variable Y . The value assigned to Y will vary from one sample point to another,

www.downloadslide.com 76 Chapter 2 Probability FIGURE 2.14 Partitioning S into subsets that define the events Y = 0, 1, 2, 3, and 4 0 4 2 1 3 S but some points may be assigned the same numerical value. Thus, we have defined a variable that is a function of the sample points in S, and {all sample points where Y = a} is the numerical event assigned the number a. Indeed, the sample space S can be partitioned into subsets so that points **with**in a subset are all assigned the same value of Y . These subsets are mutually exclusive since no point is assigned two different numerical values. The partitioning of S is symbolically indicated in Figure 2.14 for a random variable that can assume values 0, 1, 2, 3, and 4. DEFINITION 2.12 A random variable is a real-valued function for which the domain is a sample space. EXAMPLE 2.24 Solution Define an experiment as tossing two coins and observing the results. Let Y equal the number of heads obtained. Identify the sample points in S, assign a value of Y to each sample point, and identify the sample points associated **with** each value of the random variable Y . Let H and T represent head and tail, respectively; and let an ordered pair of symbols identify the outcome for the first and second coins. (Thus, HT implies a head on the first coin and a tail on the second.) Then the four sample points in S are E 1 : HH, E 2 : HT, E 3 : TH and E 4 : TT. The values of Y assigned to the sample points depend on the number of heads associated **with** each point. For E 1 : HH, two heads were observed, and E 1 is assigned the value Y = 2. Similarly, we assign the values Y = 1 to E 2 and E 3 and Y = 0toE 4 . Summarizing, the random variable Y can take three values, Y = 0, 1, and 2, which are events defined by specific collections of sample points: {Y = 0} ={E 4 }, {Y = 1} ={E 2 , E 3 }, {Y = 2} ={E 1 }. Let y denote an observed value of the random variable Y . Then P(Y = y) is the sum of the probabilities of the sample points that are assigned the value y.

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714 Chapter 14 Analysis of Categori

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742 Chapter 15 Nonparametric Statis

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