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Overview of Probability

Overview of Probability. Definitions of Probability. Outcome : perform an experiment and obtaining an outcome Event A : is a particular outcome Sample space : all of possible outcomes. Relative Frequency Definition. Perform an experiment a number of times; each time is called a trial

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Overview of Probability

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  1. Overview of Probability

  2. Definitions of Probability • Outcome: perform an experiment and obtaining an outcome • Event A: is a particular outcome • Sample space: all of possible outcomes

  3. Relative Frequency Definition • Perform an experiment a number of times; each time is called a trial • For each trial, observe whether the event A occurs where n is the number of trials nA is the number of occurrences of A

  4. Set Theory Preliminaries • A set is a collection of things. • The things that together make up the set are elements. • A universal set is the set of all things that we could possibly consider in a given context. • We will use the letter S to denote the universal set. • The null set, by definition, has no elements. • The notation for the null set is φ. • By definition φ is a subset of every set. • For any set A, φ ⊂ A.

  5. Venn Diagrams

  6. Example: the roll of a die. TheSample space is S={1,2,3,4,5,6} • The six outcomes are the sample points of the experiment. • An event is a subsetof S, and may consist of any number of sample points. For example: A={2,4} • The complementof the event A, denoted by , consists of all the sample points in S that are not in A:

  7. Two events are said to be mutually exclusiveif they have no sample points in common – that is, if the occurrence of one event excludes the occurrence of the other. For example: A={1,2} B={1,3,6} A and B are mutually exclusive events • The union of two events in an event that consists of all the sample points in the two events. For example: C={1,2,3} D=B∪C={1,2,3,6} A∪ =S • The intersection of two events is an event that consists of the points that are common to the two events. For example: E=B∩C={1,3} • When the events are mutually exclusive, the intersection is the null event, denoted as φ. For example: A∩ = φ

  8. 0 ≦Pr[A] ≦1 for all of event A • Pr[]=1 • Pr[A ∪B]=Pr[A]+Pr[B] if A and B are mutually exclusive • Pr[ ]=a=Pr[A] • Pr[A ∩B]=0if A and B are mutually exclusive • Pr[A ∪B]=Pr[A]+Pr[B]- Pr[A ∩B] • Pr[A ∪B ∪C]=Pr[A]+Pr[B]+ Pr[C]-Pr[A∩B]- Pr[B∩C]-Pr[A∩C]+ Pr[A∩B∩C]

  9. Associated with each event A contained in S is its probability P(A). • Three postulations: • P(A)≥0. • The probability of the sample space is P(S)=1. • Suppose that Ai ,i=1, 2, …, are a number of events in the sample space S such that Then the probability of the union of these mutually exclusive events satisfies the condition:

  10. Joint events and joint probabilities • If one experiment has the possible outcomes Ai , i=1,2,…,n, and the second experiment has the possible outcomesBj , j=1,2,…,m , then the combined experiment has the possible joint outcomes (Ai , Bj ) i=1,2,…,n , j=1,2,…,m • Associated with each joint outcome (Ai , Bj ) is the joint probability P (Ai , Bj ) which satisfies the condition: 0≦P (Ai , Bj ) ≦1 • 兩個或兩個以上事件聯合發生時，稱為組合機率(joint probability or compound probability)。 Pr(A∩B)＝Pr(A) ×Pr(B│A)＝Pr(B) ×Pr(A│B) but A and B are independent（獨立）if and only if : Pr(A∩B) ＝ Pr(A) ×Pr(B) • Assuming that the outcomes Bj , j =1,2,…,m, are mutually exclusive, it follows that: • If all the outcomes of the two experiments are mutually exclusive,then:

  11. Conditional probabilities • The conditional probability of the event A given the occurrence of the event B is defined as: provided P(B)>0. • The probability of the event B conditioned on the occurrence of the event A is defined as: provided P(A)>0 • P(A,B) = P(A | B)P(B) = P(B | A)P(A) • P(A, B) is interpreted as the probability of A∩B That is, P(A, B) denotes the simultaneous occurrence of A and B. • If two events and are mutually exclusive, A∩B =Φ, then P(A|B)=0. • If B is a subset of A, we have A∩B=B and P(A|B)=1

  12. Bayes’ theorem • If Ai , i=1,2,…,n are mutually exclusive events such that and is an arbitrary event with nonzero probability, then • P(Ai) represents their a priori probabilities and P(Ai |B) is the a posteriori probability of Aiconditioned on having observed the received signal B. A1 A2 B A3 A4

  13. Statistical independence : • If the occurrence of does not depend on the occurrence of B, then P(A | B)=P(A) • P(A,B) = P(A | B)P(B) = P(A)P(B) • When the events A and B satisfy the relation P(A,B) = P(A)P(B), they are said to be statistically independent. • Three statistically independent events A1, A2, and A3must satisfy the following conditions: P(A1, A3 ) = P(A1)P(A3 ), P(A1, A2) = P(A1)P(A2), P(A2, A3 ) = P(A2)P(A3 ), P(A1, A2, A3 ) = P(A1)P(A2) P(A3 ),

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