Sample Spaces Probability Addition Rules Conditional Pro. Multi. Rules Total Pro. Rule Independence Bayes’Theorem Summary 1/38 15/06/13 Department of Mathematics PROBABILITY & STATISTICS Learning objectives 1. Sample Spaces and Events 2. Interpretations of Probability 3. Addition Rules 4. Conditional Probability 5. Multiplication and Total Probability Rules 6. Independence 7. Bayes’ Theorem 1. Sample Spaces and Events 2. Interpretations of Probability 3. Addition Rules 4. Conditional Probability 5. Multiplication and Total Probability Rules 6. Independence 7. Bayes’ Theorem Chapter 2: Probability Sample Spaces Probability Addition Rules Conditional Pro. Multi. Rules Total Pro. Rule Independence Bayes’Theorem Summary 2/38 15/06/13 Department of Mathematics Sample spaces and events Sample Spaces Definition Random experiment Definition Random experiment • An experiment that can result in different outcomes, even though it is repeated in the same manner every time, is called a random experiment. • The set of all possible outcomes of a random experiment is called the sample space of the experiment. The sample space is denoted as S. • An event is a subset of the sample space of a random experiment. • An experiment that can result in different outcomes, even though it is repeated in the same manner every time, is called a random experiment. • The set of all possible outcomes of a random experiment is called the sample space of the experiment. The sample space is denoted as S. • An event is a subset of the sample space of a random experiment. Sample Spaces Probability Addition Rules Conditional Pro. Multi. Rules Total Pro. Rule Independence Bayes’Theorem Summary 3/38 15/06/13 Department of Mathematics Sample spaces and events Example Example Random experiment: Roll a die Sample space: S ={1, 2, 3, 4, 5, 6} Event: E 1 = {Die is even}={2, 4, 6} E 2 = {Die is odd}={1, 3, 5} Random experiment: Roll a die Sample space: S ={1, 2, 3, 4, 5, 6} Event: E 1 = {Die is even}={2, 4, 6} E 2 = {Die is odd}={1, 3, 5} Sample spaces Sample Spaces Probability Addition Rules Conditional Pro. Multi. Rules Total Pro. Rule Independence Bayes’Theorem Summary 4/38 15/06/13 Department of Mathematics Sample spaces and events Tree Diagrams Sample spaces can also be described graphically with tree diagrams. – When a sample space can be constructed in several steps or stages, we can represent each of the n 1 ways of completing the first step as a branch of a tree. – Each of the ways of completing the second step can be represented as n 2 branches starting from the ends of the original branches, and so forth. Sample spaces Sample Spaces Probability Addition Rules Conditional Pro. Multi. Rules Total Pro. Rule Independence Bayes’Theorem Summary 5/38 15/06/13 Department of Mathematics Sample spaces and events Example Example A probability experiment consists of tossing a coin and then rolling a six-sided die. Describe the sample space. A probability experiment consists of tossing a coin and then rolling a six-sided die. Describe the sample space. Tree diagram: H1 H2 H3 H4 H5 H6 T1 T2 T3 T4 T5 T6 The sample space has 12 outcomes: S = {H1, H2, H3, H4, H5, H6, T1, T2, T3, T4, T5, T6} Sample spaces Sample Spaces Probability Addition Rules Conditional Pro. Multi. Rules Total Pro. Rule Independence Bayes’Theorem Summary 6/38 15/06/13 Department of Mathematics Sample spaces and events Example Example Each message in a digital communication system is classified as to whether it is received within the time specified by the system design. If three messages are classified, use a tree diagram to represent the sample space of possible outcomes. Each message in a digital communication system is classified as to whether it is received within the time specified by the system design. If three messages are classified, use a tree diagram to represent the sample space of possible outcomes. Sample spaces Sample Spaces Probability Addition Rules Conditional Pro. Multi. Rules Total Pro. Rule Independence Bayes’Theorem Summary 7/38 15/06/13 Department of Mathematics Sample spaces and events Basic Set Operations The union of two events is the event that consists of all outcomes that are contained in either of the two events. We denote the union as E 1 ∪E 2 . The intersection of two events is the event that consists of all outcomes that are contained in both of the two events. We denote the intersection as E 1 ∩E 2 . The complement of an event in a sample space is the set of outcomes in the sample space that are not in the event. We denote the component of the event E as E’. Sample Spaces Sample Spaces Probability Addition Rules Conditional Pro. Multi. Rules Total Pro. Rule Independence Bayes’Theorem Summary 8/38 15/06/13 Department of Mathematics Sample spaces and events Venn Diagrams mutually exclusive Sample Spaces Sample Spaces Probability Addition Rules Conditional Pro. Multi. Rules Total Pro. Rule Independence Bayes’Theorem Summary 9/38 15/06/13 Department of Mathematics Sample spaces and events Sample Spaces A ∪ (B ∪ C) = (A ∪ B) ∪ C A ∩ (B ∩ C) = (A ∩ B) ∩ C A ∩ (B ∪ C) = (A ∩ B) ∪ (A ∩ C) (A ∪ B)’ = A’ ∩ B’ (A ∩ B)’ = A’ ∪ B’ A = (A ∩ B) ∪ (A ∩ B’) Important properties: Important properties: Sample Spaces Probability Addition Rules Conditional Pro. Multi. Rules Total Pro. Rule Independence Bayes’Theorem Summary 10/38 15/06/13 Department of Mathematics Interpretations of Probability There are three approaches to assessing the probability of an uncertain event: 1. a priori classical probability: the probability of an event is based on prior knowledge of the process involved. 2. empirical classical probability: the probability of an event is based on observed data. 3. subjective probability: the probability of an event is determined by an individual, based on that person’s past experience, personal opinion, and/or analysis of a particular situation. Introduction Probability [...]... outcomes Total Pro Rule Probability of Occurrence = Independence 2 empirical classical probability Bayes’Theorem Summary 15/06/13 Probability of Occurrence = number of favorable outcomes observed total number of outcomes observed Department of Mathematics 11/38 Interpretations of Probability Sample Spaces Probability Probability Addition Rules Example priori classical probability Find the probability of selecting... Rule Independence Bayes’Theorem Probability of Face Card = X number of face cards = T total number of cards X 12 face cards 3 = = T 52 total cards 13 Summary 15/06/13 Department of Mathematics 12/38 Interpretations of Probability Sample Spaces Probability Probability Addition Rules Example empirical classical probability Find the probability of selecting a male taking statistics from the population... Probability Sample Spaces Probability Addition Rules Conditional Pro Conditional Pro Multi Rules Total Pro Rule Independence Bayes’Theorem Summary 15/06/13 Department of Mathematics 20/38 Conditional Probability Sample Spaces Probability Addition Rules Definition Conditional Probability Definition Conditional Probability The conditional probability of an event B given an event The conditional probability of an... Independence Bayes’Theorem Summary 15/06/13 number of males taking stats total number of people 84 = = 0.191 439 Probability of Male Taking Stats = Department of Mathematics 13/38 Interpretations of Probability Sample Spaces Probability Probability Addition Rules Conditional Pro Axioms of Probability Probability is a number that is assigned to each member of a collection of events from a random experiment...Interpretations of Probability Sample Spaces Probability Probability Addition Rules Conditional Pro Equally Likely Outcomes Whenever aasample space consists of N possible outcomes that are Whenever sample space consists of N possible outcomes that are equally likely, the probability of each outcome is 1/N equally likely, the probability of each outcome is 1/N 1 a priori classical probability Multi Rules... conditional probability of D given F, and it is interpreted as the probability that a part is defective, given that the part has a surface flaw Department of Mathematics 18/38 Conditional Probability Sample Spaces Example 2-22(page 42) Probability Addition Rules Conditional Pro Conditional Pro Multi Rules Total Pro Rule Independence Bayes’Theorem Summary 15/06/13 Department of Mathematics 19/38 Conditional Probability. .. Conditional Probability Sample Spaces Conditional Probability Probability Addition Rules To introduce conditional probability, consider an example involving manufactured parts Conditional Pro Conditional Pro Multi Rules Let D denote the event that a part is defective and let F denote the event that a part has a surface flaw Total Pro Rule Independence Bayes’Theorem Summary 15/06/13 Then, we denote the probability. .. Rule Probability Addition Rules Conditional Pro Multi Rules Multi Rules Total Pro Rule Independence Bayes’Theorem Summary 15/06/13 P(A ∩ B) = P(A|B)P(B) = P(B ∩ A)P(A) Example The probability that an automobile battery subject to high engine compartment temperature suffers low charging current is 0.7 The probability that a battery is subject to high engine compartment temperature is 0.05 The probability. .. Total Probability Rule Sample Spaces Partition of an event Probability Addition Rules Conditional Pro Multi Rules Figure 2-15 Partitioning an event into two mutually exclusive subsets Total Pro Rule Total Pro Rule Figure 2-16 Partitioning an event into several mutually exclusive subsets Independence Bayes’Theorem Summary 15/06/13 Department of Mathematics 25/38 Total Probability Rule Sample Spaces Probability. .. Probability Addition Rules Conditional Pro Total Probability Rule: two events P(B) = P(BA) + P(BA’) = P(B|A)P(A) + P(B|A’)P(A’) Total Probability Rule: multiple events Multi Rules Total Pro Rule Total Pro Rule E1 ∪ E2 ∪ … Ek =S Independence Bayes’Theorem Summary 15/06/13 Department of Mathematics 26/38 Total Probability Rule Sample Spaces Example 2-27 (page 48) Probability Addition Rules Conditional Pro . Interpretations of Probability Example empirical classical probability Example empirical classical probability Find the probability of selecting a male taking statistics. Interpretations of Probability There are three approaches to assessing the probability of an uncertain event: 1. a priori classical probability: the probability