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Announcements Organization of the course β 4 quizzes, 5* homeworks, and Final Important Dates on Syllabus sheet Textbook β Probability and Statistics for Engineers and Scientists, Anthony Hayter, 4th edition, Lecture Notes by Dr. Sarath Jagupilla HW # 1 problems β 1.2.11, 1.3.14, 1.4.10, 1.5.18, 1.6.10, 1.7.8, 1.7.18 HW#1 Due on February 2, 2015 1. Probability Probability is a measure of confidence that an event will occur. The probability of an event A is denoted as P(A). The complement of an event A is the collection of all outcomes that are not contained in event A. It is denoted by Ac. π(π΄π ) = 1 β π(π΄) An event is any collection of outcomes of an experiment. An experiment is any process that generates more than one outcome. The collection of all possible outcomes of an experiment is the sample space for that experiment. Sets and Axioms of Probability Probability, as are many other fields in mathematics, is based on set theory. Given the context of an experiment, a set is any collection of its outcomes. Venn diagrams, conceived by John Venn in 1880, are great tools to learn simple relationships between sets. A Venn diagram usually consists of a rectangle which represents the sample space, and any closed shape within the rectangle which represents an event. Figure 1-1 β Venn Diagram From the Venn diagram it is clear that the event is always a subset of the sample space. The two extremes are that it can either occupy the entire sample space or the event has no possible occurrences in the sample space. These observations are captured in the following axioms of probability. 1) The probability of the sample space, S, is one i.e., P(S) = 1 2) The probability of an event that has no possible occurrences is zero i.e., π(β ) = 0 3) The probability of any event A always lies between zero and one i.e., 0 β€ π(π΄) β€ 1 Unions and Intersections Some basic relationships could be developed between two sets using Venn diagrams. Consider two sets representing the events A and B as shown, Figure 1-2 β Unions and Intersections The intersection of the two events A and B is the collection of all outcomes that occur in both the events simultaneously. The probability of the intersection is denoted by π(π΄ β© π΅). The union of the two events A and B is the collection of all outcomes that occur in at least one of the two events. The probability of the union is denoted by π(π΄ βͺ π΅). The relationship between the union and intersection of two events is given by, π(π΄ βͺ π΅) = π(π΄) + π(π΅) β ππ΄ β© π΅) We subtract the intersection once because we double counted it when we added the probabilities of A and B. Mutually Exclusive and Exhaustive Events Two events are said to be mutually exclusive if they donβt have any common elements. This implies that their intersection, π΄ β© π΅, is empty and has no events. So, two events are mutually exclusive if, π(π΄ β© π΅) = 0 Figure 1-3 β Mutually Exclusive Events Further, if two events are mutually exclusive their union is just the sum of individual probabilities, because the probability of the intersection is zero. π(π΄ βͺ π΅) = π(π΄) + π(π΅) This could be extended to any number of events, π(π΄1 βͺ π΄2 βͺ β¦ βͺ π΄π ) = π(π΄1 ) + π(π΄2 ) + β― + π(π΄π ) Two events are said to be exhaustive if, together, they occupy the entire sample space. So, two events are exhaustive if, π(π΄ βͺ π΅) = 1 Figure 1-4 β Exhaustive Events This can be extended to any number of events. So events A1, A2, β¦ , An are exhaustive if, π(π΄1 βͺ π΄2 βͺ β¦ βͺ π΄π ) = 1 Short Review β Probability, Experiment, Event, Complement of an Event, Axioms of Probability, Unions and Intersections (relation between them), Mutually Exclusive Events, Exhaustive Events. Conditional Probability, Independence, Bayeβs Rule, and Theorem of Total Probability Sometimes, we come to know that an event has already happened and we want to incorporate this fact into our probability calculations. Given that event B has already happened, the probability of A is termed conditional probability of A given B and it is calculated as, π(π΄/π΅) = π(π΄ β© π΅) π(π΅) Figure 1-5 β Conditional Probability The formula could be understood by examining the Venn diagram. Similarly, the conditional probability of event B given that A has occurred is given by, π(π΅/π΄) = π(π΄ β© π΅) π(π΄) The two equations can be combined and rearranged to result in the Bayeβs rule, π(π΄/π΅) = π(π΅/π΄). π(π΄) π(π΅) Two events are said to be independent of each other if the occurrence of either event does not affect the occurrence of the other event. So, whether event B happens or not, it would not affect the probability of event A. Therefore if A is independent of B then, π(π΄/π΅) = π(π΄) Using the conditional probability rule in the definition of independence, a condition for independence is given by, π(π΄ β© π΅) = π(π΄). π(π΅)