2-8 RANDOM VARIABLES

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2-119. The following table lists the history of 940 orders for features in an entry-level computer product. extra memory no yes 514 68 112 246

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each device is as shown. What is the probability that the circuit does not operate

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optional highspeed processor

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no yes

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Let A be the event that an order requests the optional highspeed processor, and let B be the event that an order requests extra memory. Determine the following probabilities: (a) P1A B2 (b) P1A B2 (c) P1A B2 (d) P1A B 2 (e) What is the probability that an order requests an optional high-speed processor given that the order requests extra memory (f) What is the probability that an order requests extra memory given that the order requests an optional high-speed processor 2-120. The alignment between the magnetic tape and head in a magnetic tape storage system affects the performance of the system. Suppose that 10% of the read operations are degraded by skewed alignments, 5% of the read operations are degraded by off-center alignments, and the remaining read operations are properly aligned. The probability of a read error is 0.01 from a skewed alignment, 0.02 from an off-center alignment, and 0.001 from a proper alignment. (a) What is the probability of a read error (b) If a read error occurs, what is the probability that it is due to a skewed alignment 2-121. The following circuit operates if and only if there is a path of functional devices from left to right. Assume that devices fail independently and that the probability of failure of

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2-122. A company that tracks the use of its web site determined that the more pages a visitor views, the more likely the visitor is to provide contact information. Use the following tables to answer the questions: Number of pages viewed: Percentage of visitors: Percentage of visitors in each page-view catgory that provide contact information: 1 40 2 30 3 20 4 or more 10

(a) What is the probability that a visitor to the web site provides contact information (b) If a visitor provides contact information, what is the probability that the visitor viewed four or more pages

CHAPTER 2 PROBABILITY

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MIND-EXPANDING EXERCISES

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2-123. The alignment between the magnetic tape and head in a magnetic tape storage system affects the performance of the system. Suppose that 10% of the read operations are degraded by skewed alignments, 5% by off-center alignments, 1% by both skewness and offcenter, and the remaining read operations are properly aligned. The probability of a read error is 0.01 from a skewed alignment, 0.02 from an off-center alignment, 0.06 from both conditions, and 0.001 from a proper alignment. What is the probability of a read error. 2-124. Suppose that a lot of washers is large enough that it can be assumed that the sampling is done with replacement. Assume that 60% of the washers exceed the target thickness. (a) What is the minimum number of washers that need to be selected so that the probability that all the washers are thinner than the target is less than 0.10 (b) What is the minimum number of washers that need to be selected so that the probability that one or more washers are thicker than the target is at least 0.90 2-125. A biotechnology manufacturing rm can produce diagnostic test kits at a cost of $20. Each kit for which there is a demand in the week of production can be sold for $100. However, the half-life of components in the kit requires the kit to be scrapped if it is not sold in the week of production. The cost of scrapping the kit is $5. The weekly demand is summarized as follows: weekly demand Number of units 0 50 100 200 Probability of demand 0.05 0.4 0.3 0.25 How many kits should be produced each week to maximize the mean earnings of the rm 2-126. Assume the following characteristics of the inspection process in Exercise 2-107. If an operator checks a bolt, the probability that an incorrectly torqued bolt is identi ed is 0.95. If a checked bolt is correctly torqued, the operator s conclusion is always correct. What is the probability that at least one bolt in the sample of four is identi ed as being incorrectly torqued 2-127. If the events A and B are independent, show that A and B are independent. 2-128. Suppose that a table of part counts is generalized as follows: conforms yes no supplier 1 ka kb 2 a b where a, b, and k are positive integers. Let A denote the event that a part is from supplier 1 and let B denote the event that a part conforms to speci cations. Show that A and B are independent events. This exercise illustrates the result that whenever the rows of a table (with r rows and c columns) are proportional, an event de ned by a row category and an event de ned by a column category are independent.

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IMPORTANT TERMS AND CONCEPTS In the E-book, click on any term or concept below to go to that subject. Addition rule Axioms of probability Bayes theorem Conditional probability Equally likely outcomes Event Independence Multiplication rule Mutually exclusive events Outcome Random experiment Random variables discrete and continuous Sample spaces discrete and continuous Total probability rule With or without replacement CD MATERIAL Permutation Combination

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2-1.4

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Counting Techniques (CD Only)

In many of the examples in 2, it is easy to determine the number of outcomes in each event. In more complicated examples, determining the number of outcomes that comprise the sample space (or an event) becomes more dif cult. To associate probabilities with events, it is important to know the number of outcomes both in an event and in the sample space. Some simple rules can be used to simplify the calculations. In Example 2-4, an automobile manufacturer provides vehicles equipped with selected options. Each vehicle is ordered With or without an automatic transmission With or without air conditioning With one of three choices of a stereo system With one of four exterior colors The tree diagram in Fig. 2-6 describes the sample space of all possible vehicle types. The size of the sample space equals the number of branches in the last level of the tree and this quantity equals 2 2 3 4 = 48. This leads to the following useful result.

Multiplication Rule (for counting techniques)

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If an operation can be described as a sequence of k steps, and if the number of ways of completing step 1 is n1, and if the number of ways of completing step 2 is n2 for each way of completing step 1, and if the number of ways of completing step 3 is n3 for each way of completing step 2, and so forth, the total number of ways of completing the operation is n1 n2 p nk

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EXAMPLE S2-1

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In the design of a casing for a gear housing, we can use four different types of fasteners, three different bolt lengths, and three different bolt locations. From the multiplication rule, 4 3 3 36 different designs are possible. Permutations Another useful calculation is the number of ordered sequences of the elements of a set. Consider a set of elements, such as S {a, b, c}. A permutation of the elements is an ordered sequence of the elements. For example, abc, acb, bac, bca, cab, and cba are all of the permutations of the elements of S.

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The number of permutations of n different elements is n! where n! n 1n 12 1n 22 p 2 1 (S2-1)

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This result follows from the multiplication rule. A permutation can be constructed by selecting the element to be placed in the rst position of the sequence from the n elements, then selecting the element for the second position from the n 1 remaining elements, then selecting the element for the third position from the remaining n 2 elements, and so forth. Permutations such as these are sometimes referred to as linear permutations. In some situations, we are interested in the number of arrangements of only some of the elements of a set. The following result also follows from the multiplication rule.

The number of permutations of a subset of r elements selected from a set of n different elements is P rn n 1n 12 1n 22 p 1n r 12 1n n! r2! (S2-2)

EXAMPLE S2-2

A printed circuit board has eight different locations in which a component can be placed. If four different components are to be placed on the board, how many different designs are possible Each design consists of selecting a location from the eight locations for the rst component, a location from the remaining seven for the second component, a location from the remaining six for the third component, and a location from the remaining ve for the fourth component. Therefore,

8 P4

8! 4!