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Chapter2_print - STAT511 Summer 2009 Lecture Notes 1...

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STAT511 — Summer 2009 Lecture Notes 1 Chapter 2: Events and Laws of Probability June 17, 2009 2 Probability Chapter Overview This chapter focuses on “Probability” Random Experiments, Sample Space and Events. Random Experiment (R.E.) Sample Space Events Examples Introduction to Probability Axiom Properties Examples Counting Techniques Combinations Permutations Conditional Probability Definition Related rules and theorem Independence 2.1 Random Experiment, Sample Space & Events Random Experiment, Sample Space & Events Random Experiment (R.E.) A R.E. is any action or process whose outcome is subject to uncertainty; i.e. many outcomes, each has a certain chance to happen. Sample Space of a R.E. Denoted S , is the set of all possible outcomes of the R.E. Event of a R.E. Denoted with capital letters (event A , B , E 1 , E 2 , etc), an event is any collection (subset) of outcomes contained in the sample space S of the R.E. An event is said to be simple if it contains of exactly one outcome. An event is said to be compound if it contains more than one outcome. Purdue University Chapter2˙print.tex; Last Modified: June 17, 2009 (W. Sharabati)
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STAT511 — Summer 2009 Lecture Notes 2 R.E. Example 2.1.1 Example 1 (Roll a Die) . What is the sample space? Outcomes: Landing 1, 2, 3, 4, 5, 6 face up. Sample Space: S = { 1 , 2 , 3 , 4 , 5 , 6 } . Events: Let E 1 be the event of getting a 2, then E 1 = { 2 } (simple event). Let E 2 be the event of getting a number greater than 3, then E 2 = { 4 , 5 , 6 } (compound event). Exercise 1. Let E 3 be the event of rolling a seven on a single die, what is E 3 ? 2. Let E 4 be the event of rolling an even number on a die, what is E 4 ? Example 2 (2.1.2 Tossing a Coin) . Toss a fair coin until a head appears, what is the sample space? Sample Space: S = { H, TH, TTH, TTTH, TTTTH, · · · } . Events: 1. Let A be the event of tossing exactly 2 tosses, then A = { TH } . (simple event) 2. Let B be the event of tossing less than 4 tosses, then B = { H, TH, TTH } (compound event). 3. Let C be the event of tossing 2 and 5 tosses, then C = { TH, TTH, TTTH, TTTTH } (compound event). 4. Let D be the event of tossing more than 3 tosses, then D = { TTTH, TTTTH, TTTTTH, · · · } (compound event). Relations From Set Theory An event is a set of outcomes, we may use set theory to form new events . Union A B : is the event containing all outcomes in A or B . Intersection A B : is the event containing all outcomes in both A and B . Compliment A c : is the event containing all outcomes in the sample space S that are not contained in A . A A c = S Mutually Exclusive (Disjoint) : A and B are said to be mutually exclusive if they contain no outcome in common. A B = φ φ is the empty set. Disjoint Events Example 3 (Drawing Cards) . When drawing a single card from a standard deck of cards, if event A = { heart, diamond } ( red ) and event B = { spade, club } (black), then A and B are mutually exclusive.
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