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 1 –Example 2: Seating ArrangementsExample 2 Permutation: A permutation of a set of objects is a listing of the objects in some specified order…

3. Permutations Example 3: Batting OrdersExample 3 Example 4: Beauty PageantExample 4 Example 5: License PlatesExample 5 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 ShirtsExample 6 Definition of Combinations Example 7: ComputeExample 7 Example 8: Five Shirts revisitedExample 8 Example 9: CommitteeExample 9 Example 10: Officers of Committee…Example 10

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 SpaceExample 11 Definition: An event is any subset of the sample space. Example 12: Some Events of Single DieExample 12 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 DieExample 13 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 dieExample 14 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…Example 15

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

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