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7Some of the details mentioned in this section were derived from studying Sigaba software simulators, so it is possible that there are minor discrepancies with the way that an actual Sigaba machine operates. However, none of these details appear t o have any significant effect on the analysis or attack discussed here.

14-1 Introduction 506 14-2 Some Applications of Designed Experiments (CD Only) 506 14-3 Factorial Experiments 506 14-4 Two-Factor Factorial Experiments 510

14-4.1 Statistical Analysis of the FixedEffects Model 511 14-4.2 Model Adequacy Checking 517 14-4.3 One Observation Per Cell 517 14-4.4 Factorial Experiments with Random Factors: Overview 518

;stepping:

Probability is a number that is assigned to each member of a collection of events from a random experiment that satis es the following properties: If S is the sample space and E is any event in a random experiment, (1) P1S2 1

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The property that 0 P1E2 1 is equivalent to the requirement that a relative frequency must be between 0 and 1. The property that P(S) 1 is a consequence of the fact that an outcome from the sample space occurs on every trial of an experiment. Consequently, the relative frequency of S is 1. Property 3 implies that if the events E1 and E2 have no outcomes in common, the relative frequency of outcomes in E1 E2 is the sum of the relative frequencies of the outcomes in E1 and E2. These axioms imply the following results. The derivations are left as exercises at the end of this section. Now, P1 2 and for any event E, P1E 2 1 P1E2 0

For example, if the probability of the event E is 0.4, our interpretation of relative frequency implies that the probability of E is 0.6. Furthermore, if the event E1 is contained in the event E2, P1E1 2 EXERCISES FOR SECTION 2-2

As with the Enigma rotors, there are 26! choices for each of the cipher and control rotors. Similarly, there are lo! choices for each of the index rotors. This gives a total keyspace of about

2-34. Each of the possible ve outcomes of a random experiment is equally likely. The sample space is {a, b, c, d, e}. Let A denote the event {a, b}, and let B denote the event {c, d, e}. Determine the following: (a) P1A2 (b) P1B2 (c) P1A 2 (d) P1A B2 (e) P1A B2 2-35. The sample space of a random experiment is {a, b, c, d, e} with probabilities 0.1, 0.1, 0.2, 0.4, and 0.2, respectively. Let A denote the event {a, b, c}, and let B denote the event {c, d, e}. Determine the following: (a) P1A2 (b) P1B2 (c) P1A 2 (d) P1A B2 (e) P1A B2 2-36. A part selected for testing is equally likely to have been produced on any one of six cutting tools. (a) What is the sample space (b) What is the probability that the part is from tool 1 (c) What is the probability that the part is from tool 3 or tool 5 (d) What is the probability that the part is not from tool 4 2-37. An injection-molded part is equally likely to be obtained from any one of the eight cavities on a mold. (a) What is the sample space (b) What is the probability a part is from cavity 1 or 2 (c) What is the probability that a part is neither from cavity 3 nor 4

2-38. A sample space contains 20 equally likely outcomes. If the probability of event A is 0.3, how many outcomes are in event A 2-39. Orders for a computer are summarized by the optional features that are requested as follows: proportion of orders 0.3 0.5 0.2

(261910. (10!)5

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