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Unformatted text preview: Introduction to Probability Enrique CamposN´ a˜nez The George Washington University ApSc 116 Table of Contents Interpretations of Probability Probability and Set Theory The Definition of Probability Counting Methods Interpretations of Probability Interpretation 1: Frequency interpretation I Probability = limit of relative frequency, i.e., P ( A ) = lim n →∞ Number of times event A occurs in n trials n I Cannot be applied when few repetitions are possible Example (Coin toss) We say the probability of a head is 1 / 2 because we expect that if repeat a coin toss many times, we expect that approximately half of the times we will obtain a head. Interpretations of Probability Interpretation 2: Equally likely outcomes I The six sides of a die are “equally likely” ⇒ each has a probability of 1/6. I How can we deal with situations where there are no equally likely outcomes? Example (Throwing two dice) When we consider each possible (ordered) combinations (i.e., (1 , 2)), each is unique and has a probability of 1 / 36. Example (Sum of two dice) If we consider the sum of two dice, we have an experiment whose outcomes are not equally likely. Interpretations of Probability Interpretation 3: Subjective I P ( A ) = relative likelihood that a particular assigns to the occurrence of event A , usually based on experience or prior knowledge. I The difficulty is that two people may assign different probabilities to the same event. I This interpretation can agree with the previous two interpretations. Example (Supply Chain Security) What is the probability that a container entering the U.S.A. carries hazardous material? The probability is unknown, but perhaps can be assessed by experts (which most certainly will not agree). Experiments and Events Definition (Experiment) An experiment is any process whose outcome is not known in advance with certainty. All possible outcomes of an experiment can be specified before the experiment is performed, we call this the sample space; we can also assign probabilities to each of the possible outcomes. Definition (Event) We define an event as any subset of the sample space. Experiments and Events Example 1: Tossing a Single Die I The face of a single die is unknown, and therefore we can consider it an experiment. I The set of possible outcomes is simply { 1 , 2 , 3 , 4 , 5 , 6 } . I Obtaining an even number is an event, i.e., a subset of the outcomes consisting of { 2 , 4 , 6 } . I Obtaining a number less than 3 is an event consisting of the set { 1 , 2 } . Experiments and Events Example 2: The Life of a Component I The duration of an electrical component is unknown in advance, and hence can be considered random....
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 Fall '08
 TomMazzuchi
 Set Theory, Probability theory, Naive set theory, Empty set

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