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3+Equally+Likely+Outcomes+Student+Version

# 3+Equally+Likely+Outcomes+Student+Version - MS&E 220...

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1 MS&E 220 Probabilistic Analysis Lecture 3 Systems with Equally Likely Outcomes Samuel S. Chiu Stanford University Chapter 3: Fundamentals and Systems with equally likely outcomes The foundation: axioms Simple rules: set algebra, trees, Venn Diagrams Model building: Possibilities & Probabilities Simple procedure for systems with equally likely outcomes – Determine all possible outcomes: count them – Determine the outcomes you want: count them – Take the ratio Chapter Concepts Axioms of Probability Theory Partitioning of The Careful (and Very Careful) Accountant The universe Adding Probabilities An Event Accounting for Double Counting Conditional Reasoning Two Components of a Probability Model Partitioning, Scenario Analysis What are Possible The pencil Test How Likely Analytical Solution: Partitioning of Globally and Potentially Complex An Event Dynamic Equation The Universe Relating Locally and Simple

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2 Axioms: in words Probability is between zero and one: a natural way to “normalize” Total probability is one: something has to happen Rules to tell you when you can add probabilities: A B A B Axioms: in notations Draw your boundary by defining a universe Ω For each subset E ⊆ Ω, define a function P ( E ): – 0 P ( E ) 1 P ( Ω ) = 1 – For any collection of mutually exclusive events: { E j } (finite or infinite collection) we can compute: [ ] ). ( = j j j j E P E P The third Axiom: Adding probabilities when there is no overlapping P (union) = sum of individual probabilities
3 A tree representation naturally does that: no overlapping Scenarios S 1 S 2 …. …. S k …. S n p n p 1 p 2 Some useful propositions P ( E ) + P ( E c ) = P ( E ) + P ( E ’) = 1 P ( Φ ) = 0 E F implies P ( E ) P ( F ). The careful accountant: when sets (events) overlap: P ( E F ) = P ( E ) + P ( F ) - P ( EF ). The careful accountant: A Venn Diagram view P ( E F ) = P ( E ) + P ( F ) - P ( EF )

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4 Generalizing: A very careful accountant P ( E F G ) = _______________________________ E F G P ( A B C D E ) Sum of the probabilities of one at a time minus the probabilities of (the intersection) of two at a time plus the probabilities of (the joint intersection) of three at a time minus the probabilities of (the joint intersection) of four at a time • ….. The concept of partitioning: think scenarios Of a sample space - universe E 1 E 2 E 4 E 3 E 6 E 5 E 1 E 3 E 2 E 4 E 6 E 5 P ( E 1 ) P ( E 3 ) P ( E 6 )
5 Partitioning an event into non-overlapping components Why? A 1 A 3 A 4 A 5 A 2 . 5 1 U = = j j A A Partitioning an event into non-overlapping components Because we can add the component probabilities A 1 A 3 A 4 A 5 A 2 . 5 1 U = = j j A A [ ] ). ( ) ( 5 1 5 1 = = = = j j j j A P A P A P U Examples

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6 Birthday matches Find probability that – there is no birthday matches in a room of n – Or its complement: at least two people with the same birthday Which one is easier?
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3+Equally+Likely+Outcomes+Student+Version - MS&E 220...

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