Download presentation
Presentation is loading. Please wait.
Published byGeorgiana Alexander Modified over 9 years ago
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
Similar presentations
© 2025 SlidePlayer.com. Inc.
All rights reserved.