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Unformatted text preview: Math 211 Introduction to Statistics Chapter 5 Probability and Some Probability Distributions Key words : Sample space, sample point, tree diagram, events, complement, union and intersection of an event, mutually exclusive events; Counting techniques: multiplication rule, permutation, combination, probability of an event; Additive rules; Conditional probability, independent events, multiplicative rules, Bayes’ rule. SAMPLE SPACE Sample Space. The set of all possible outcomes of a statistical experiment is called a Sample space. It is represented by the symbol S. Sample Point. Each outcome in a sample space is called an element or a sample point of the sample space. Example 1. (a) Consider the experiment of tossing a coin. The sample space S of possible outcomes may be written as S = { } , H T . (b) Consider the experiment of flipping a die. Then the elements of the sample space S is listed as S = { } 1,2,3,4,5,6 . (c) Now consider the experiment of tossing a die and then a coin once. The resultant sample space can be obtained using TREE DIAGRAM. That is, Therefore the sample sample S is S = { } 1 ,1 ,2 ,2 ,3 ,3 ,4 ,4 ,5 ,5 ,6 ,6 H T H T H T H T H T H T . Sonuc Zorlu Lecture Notes 1 Math 211 Introduction to Statistics Event. An event is a subset of a sample space. For example { } 4 ,6 A H T = is an event defined on S . One can define 12 2 events on S. Empty set ∅ , is an impossible event and S is a sure event. Any subset of S is represented by capital letters such as A, B, C… The Complement of an Event. The complement of an event A with respect to S is the subset of all elements of S that are not in A . The complement of A is denoted by the symbol ' A or c A . The Intersection of Events. The intersection of two events A and B, denoted by the symbol A B ∩ is the event containing all elements that are common to A and B. Mutually Exclusive Events. Two events A and B are mutually exclusive or disjoint if A B ∩ = ∅ , that is if A and B have no common elements in common. The Union of Events. The union of two events A and B, denoted by the symbol A B ∪ is the event containing all elements that belong to A or B or both . Important Notes. The following results may easily be verified by means of Venn diagrams. (1) A ∩ ∅ = ∅ (2) A A ∪ ∅ = (3) ' A A ∩ = ∅ (4) ' A A S ∪ = (5) ' S = ∅ (6) ' S ∅ = (7) ( 29 ' ' A A = (8) ( 29 ( 29 ' ' ' A B A B ∩ = ∪ 1 st De Morgan Rule (9) ( 29 ' ' ' A B A B ∪ = ∩ 2 nd De Morgan Rule Example 2. If { } 0,1,2,3,4,5,6,7,8,9 S = and { } 0,2,4,6,8 A = , { } 1,3,5,7,9 B = , { } 2,3,4,5 C = and { } 1,6,7 D = , list the elements of the sets corresponding to the following events: (a) A C S ∪ = (b) A B ∩ = ∅ ( A and B are mutually exclusive events) (c) { } ' 0,1,6,7,8,9 C = (d) ( 29 { } { } ( 29 { } { } { } { } ' 0,1,6,7,8,9 1,6,7 1,3,5,7,9 1,6,7 1,3,5,7,9 1,7 C D B ∩ ∪ = ∩ ∪ = ∪ = (e) ( 29 { } ' ' 0,1,6,7,8,9 S C C ∩ = = (f) { } { } { } ' 2,4 0,2,3,4,5,8,9 2,4 A C D A C ∩ ∩ =...
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 Spring '10
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 Normal Distribution, Probability distribution, Probability theory, Sonuc Zorlu

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