2_BasicProba

# 2_BasicProba - i i i 2 Basic Probability Chapter Outline...

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i i i 2 Basic Probability Chapter Outline 2.1 Introduction 2.2 Random Experiments 2.3 Sample Space 2.4 Probability Axioms and Theorems 2.4.1 Three Probability Axioms 2.4.2 Eight Probability Theorems 2.5 Conditional Probability and Theorems 2.5.1 Conditional Probability 2.5.2 Conditional Probability Theorems 2.5.3 Multiplicative Rule and To- tal probability Theorem 2.5.4 Bayes’s Theorem 2.5.5 Independent Events 2.6 Probability Interpretation
i i i 2.1 Introduction 2.2 Random Experiments Defnition 2.1 Random Experiment: An experiment (a procedure) that can result in a different outcome each time it is performed, even though it is repeated in the same manner every time, is called a random experiment. input (controllable) processing output random factor (uncontrollable) Figure 2.1: Concept of a random experiment

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i i i Example 2.1 Flipping a fair coin n time, what does the frequency converge? Solution n probability = 0.5 frequency Figure 2.2: One realization of ±ipping a fair coin ± Example 2.2 Monty Hall problem. There are three doors: 1, 2, 3. A grand prize lies behind one door; nothing lies behind the other doors. The contestant wins the prize by choosing the correct door. The game begins with the contestant choosing an initial door. Monty Hall then opens one of the other doors, showing that nothing lies behind it, and asks the contestant whether she wants to keep the original door, or to choose the other unopened door. Suppose that this contestant decides to change doors. what does the frequency, which the contestant wins the gift, converge? Solution n probability=0.67 frequency Figure 2.3: One realization of playing Money Hall game ±
i i i 2.3 Sample Space Defnition 2.2 Sample point: A possible outcome of a random experiment is called a sample point. A sample point is denoted as s . Defnition 2.3 Sample space: A set containing all possible outcomes of a random experiment is called a sample space of the experi- ment. A sample space is denoted as S. We have “ S”. A sample space is “discrete” if it has a Fnite or countably inFnite number of members. Defnition 2.4 Event: An event is a subset of the sample space of a random experiment. Defnition 2.5 Event space: A set contains all possible events in a random experiment is called an event space. Let D be an event space; D ={ E : E S } . Defnition 2.6 Probability function, P: Probability is a function which as- sign a real value between 0 and 1 to any event in the event space. Defnition 2.7 Probability space: A space contains sample space, event space, and probability function (S, D ,P).

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i i i Example 2.3 When we say probability of “something”, what is the “something” in general? thepossibleoutcomesofanexperiment sample points an event Example 2.4 Why don’t we deFne the probability for the possible outcomes of a random experiment?
i i i Events 1.

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## This note was uploaded on 05/11/2011 for the course IEEM 9834211 taught by Professor Wang during the Spring '09 term at Tsinghua University.

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2_BasicProba - i i i 2 Basic Probability Chapter Outline...

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