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EXERCISES FOR SECTION 2-3

There are many ways to systematically mix the letters of the plaintext. For example, we can strengthen the columnar transposition cipher by allowing the permutation of columns and rows. Since two transpositions are involved, this is known as a double transposition cipher, which we briefly describe next.

2-49. If P 1A2 0.3 , P 1B2 0.2, and P 1A B2 0.1, determine the following probabilities: (a) P1A 2 (b) P1A B2 (c) P1A B2 (d) P1A B 2 (e) P3 1A B2 4 (f) P1A B2 2-50. If A, B, and C are mutually exclusive events with P 1A2 0.2, P 1B2 0.3, and P 1C 2 0.4, determine the following probabilities: (a) P1A B C2 (b) P1A B C2 (c) P1A B2 (d) P3 1A B2 C 4 (e) P1A B C 2 2-51. If A, B, and C are mutually exclusive events, is it possible for P(A) 0.3, P(B) 0.4, and P(C) 0.5 Why or why not 2-52. Disks of polycarbonate plastic from a supplier are analyzed for scratch and shock resistance. The results from 100 disks are summarized as follows: shock resistance high low 70 9 16 5 (a) If a shaft is selected at random, what is the probability that the shaft conforms to surface nish requirements (b) What is the probability that the selected shaft conforms to surface nish requirements or to roundness requirements (c) What is the probability that the selected shaft either conforms to surface nish requirements or does not conform to roundness requirements (d) What is the probability that the selected shaft conforms to both surface nish and roundness requirements 2-54. Cooking oil is produced in two main varieties: monoand polyunsaturated. Two common sources of cooking oil are corn and canola. The following table shows the number of bottles of these oils at a supermarket: type of oil canola corn 7 13 93 77

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(a) If a disk is selected at random, what is the probability that its scratch resistance is high and its shock resistance is high (b) If a disk is selected at random, what is the probability that its scratch resistance is high or its shock resistance is high (c) Consider the event that a disk has high scratch resistance and the event that a disk has high shock resistance. Are these two events mutually exclusive 2-53. The analysis of shafts for a compressor is summarized by conformance to speci cations. roundness conforms yes no 345 5 12 8

To encrypt with a double transposition cipher, we first write the plaintext into an array of a given size and then permute the rows and columns according to specified permutations. For example, suppose we write the plaintext ATTACKATDAWN into a 3 x 4 array:

(a) If a bottle of oil is selected at random, what is the probability that it belongs to the polyunsaturated category (b) What is the probability that the chosen bottle is monounsaturated canola oil 2-55. A manufacturer of front lights for automobiles tests lamps under a high humidity, high temperature environment using intensity and useful life as the responses of interest. The following table shows the performance of 130 lamps: useful life satisfactory unsatisfactory intensity satisfactory 117 3 unsatisfactory 8 2 (a) Find the probability that a randomly selected lamp will yield unsatisfactory results under any criteria. (b) The customers for these lamps demand 95% satisfactory results. Can the lamp manufacturer meet this demand 2-56. The shafts in Exercise 2-53 are further classi ed in terms of the machine tool that was used for manufacturing the shaft.

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