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Section 2.6 Probability and Expectation

Section 2.6 Probability and Expectation. Cryptanalyzing the Vigenere cipher is not a trivial process. A probabilistic method that allows one to determine the likely keyword length is the first step in breaking this cipher. In this section we cover the basic methods of counting things….

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Section 2.6 Probability and Expectation

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  1. Section 2.6 Probability and Expectation • Cryptanalyzing the Vigenere cipher is not a trivial process. • A probabilistic method that allows one to determine the likely keyword length is the first step in breaking this cipher. • In this section we cover the basic methods of counting things…

  2. Permutations • Factorial: If n is a nonnegative integer, then n factorial, denoted n!, is defined as: • n! = n(n-1)(n-2)…2*1 • Note: 0! = 1, and 1! = 1, by definition. • Example 1: Calculate 3!, 5!, 186!, 10! / 9! • Example 2: Seating Arrangements • Permutation: A permutation of a set of objects is a listing of the objects in some specified order…

  3. Permutations • Example 3: Batting Orders • Example 4: Beauty Pageant • Example 5: License Plates • Formula for permutation: If you have n things to choose from and you select k of those things, without replacement (You cannot select an item more than once), and the order matters (AB is different then BA), then P(n, k) = n! / (n – k)!...

  4. Combinations • A combination is similar to a permutation except that order does not matter. AB and BA are the same. • Example 6: Five Shirts • Definition of Combinations • Example 7: Compute • Example 8: Five Shirts revisited • Example 9: Committee • Example 10: Officers of Committee…

  5. Basic Probability • Definition: The sample space of an experiment is the set of all possible outcomes of an experiment. • Example 11: Single Die Sample Space • Definition: An event is any subset of the sample space. • Example 12: Some Events of Single Die • Definition of Probability: The probability of an event is a number between 0 and 1 that represents the chance of an event occurring. If A is an event, then P(A) = (the number of ways that event A can occur) / (total number of outcomes that occurs in the sample space)…

  6. Probability of Events • Example 13: Rolling Die • Facts about Probability: • Given the probability P of an event occurring • 0 ≤ P ≤ 1 • Given two events A and B that are mutually exclusive (A and B are separate) then • P(A or B) = P(A) + P(B) • Example 14: Roll a single die • Given the probability of an event A, then the probability of not A is: P(not A) = 1 – P(A). • Example 15: Not rolling 5…

  7. Probabilities • The sum of all the probabilities of mutually exclusive events in a sample space is equal to 1. • Example 16: Equal 1 probability • Example 17: Toss two Die…

  8. Probability of Simultaneous Events • Multiplication Principle of Probability • Example 18: Without Replacement…!

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