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Lec03 - Chapter Outline IOE/Stat 265 Fall 2009 Lecture#3...

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1 http://en.wikipedia.org/wiki/Birthday_paradox IOE/Stat 265, Fall 2009 Lecture #3: Lecture #3: Probability Concepts 2 Chapter Outline 2-1 Sample Spaces & Events Random Experiments Sample Spaces Events 2-2 Interpretations of Probability Axioms of Probability Interpretations of Probability Properties of Probability 2-3 Counting Techniques Combinations/Permutations 3 Ch. 2 - Learning Objectives 1. Understand and describe sample spaces and events for random experiments with graphs , tables , lists , or tree diagrams 2. Interpret probabilities and use probabilities of outcomes to calculate probabilities of events in discrete sample spaces . 3. Use permutations and combinations to count the number of outcomes in both events and sample spaces . 4. Calculate probabilities of joint events ( unions and intersections ) from individual event probabilities. 4 Probability factoids Branch of mathematics that goes back more than 300 years How did it get started? Games of Chance! Gambler’s dispute in 1654 between two famous French mathematicians: Blaise Pascal and Pierre de Fermat The game consisted in throwing a pair of dice 24 times; the problem was to decide whether or not to bet money on the occurrence of at least one "double six" during 24 throws
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5 2-1 Random Experiment An experiment that can result in different outcomes, even though it is repeated in the same manner every time, is called a random experiment. Examples: 7 2-1 Sample Spaces Sample Space Set of all possible outcomes from a random experiment. Examples: Two Possible Outcomes: part defective, not defective >2 Discrete Outcomes: rolling a die (1, 2, 3, 4, 5, 6) Continuous Variable Outcomes: length of desk measurements (infinite number of possible outcomes) Sample spaces with multiple components Suppose you have a twin-engine airplane. What are all the possible outcomes if each engine either fails or does not during a flight? 8 More Examples: Sample Spaces IOE Students Voters, TV Viewers Restaurants in Ann Arbor Automobiles produced at Ford’s Rouge plant 9 2-1 Events Subset of outcomes contained in the sample space of a random experiment. Simple Event – exactly one outcome Compound Event – more than one outcome
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10 Simple Event Unique outcome of an experiment Some examples of simple events: Flipping a Tail when tossing a coin Rolling a 3 when tossing a dice Measuring a length of 12.5 mm Getting an 85 on the first exam 11 Tree Diagrams Useful in depicting compound/multiple events and sample spaces 12 Set Theory Union Intersection Complement Mutual Exclusive or Disjoint Events 13 Venn Diagrams In depicting multiple events, Venn diagrams are excellent visual tools. A B
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14 Venn Diagram Example 1 A = Cars with Sunroofs B = Cars with Air Conditioning What does the shaded area represent ? A B 15 Venn Diagram Example 2 A = Cars with Sunroofs B = Cars with Air Conditioning What does the shaded area represent ?
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