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Lecture Discrete mathematics and its applications - Lecture 10

In this lesson you will learn about the types of operating systems and the services they provide. Then you will learn about some of the enhancements you can make to your OS using utility software. | Discrete Probability Chapter 7 With Question/Answer Animations Copyright © McGraw-Hill Education. All rights reserved. No reproduction or distribution without the prior written consent of McGraw-Hill Education. Chapter Summary Introduction to Discrete Probability Probability Theory Bayes’ Theorem Expected Value and Variance An Introduction to Discrete Probability Section 7.1 Section Summary Finite Probability Probabilities of Complements and Unions of Events Probabilistic Reasoning Probability of an Event We first study Pierre-Simon Laplace’s classical theory of probability, which he introduced in the 18th century, when he analyzed games of chance. We first define these key terms: An experiment is a procedure that yields one of a given set of possible outcomes. The sample space of the experiment is the set of possible outcomes. An event is a subset of the sample space. Here is how Laplace defined the probability of an event: Definition: If S is a finite sample space of equally likely outcomes, and E is an event, that is, a subset of S, then the probability of E is p(E) = |E|/|S|. For every event E, we have 0 ≤ p(E) ≤ 1. This follows directly from the definition because 0 ≤ p(E) = |E|/|S| ≤ |S|/|S| ≤ 1, since 0 ≤ |E| ≤ |S|. Pierre-Simon Laplace (1749-1827) Applying Laplace’s Definition Example: An urn contains four blue balls and five red balls. What is the probability that a ball chosen from the urn is blue? Solution: The probability that the ball is chosen is 4/9 since there are nine possible outcomes, and four of these produce a blue ball. Example: What is the probability that when two dice are rolled, the sum of the numbers on the two dice is 7? Solution: By the product rule there are 62 = 36 possible outcomes. Six of these sum to 7. Hence, the probability of obtaining a 7 is 6/36 = 1/6. Applying Laplace’s Definition Example: In a lottery, a player wins a large prize when they pick four digits that match, in correct order, four digits selected by a random . | Discrete Probability Chapter 7 With Question/Answer Animations Copyright © McGraw-Hill Education. All rights reserved. No reproduction or distribution without the prior written consent of McGraw-Hill Education. Chapter Summary Introduction to Discrete Probability Probability Theory Bayes’ Theorem Expected Value and Variance An Introduction to Discrete Probability Section 7.1 Section Summary Finite Probability Probabilities of Complements and Unions of Events Probabilistic Reasoning Probability of an Event We first study Pierre-Simon Laplace’s classical theory of probability, which he introduced in the 18th century, when he analyzed games of chance. We first define these key terms: An experiment is a procedure that yields one of a given set of possible outcomes. The sample space of the experiment is the set of possible outcomes. An event is a subset of the sample space. Here is how Laplace defined the probability of an event: Definition: If S is a finite sample space of equally likely