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2+Chapter+2+Student+Version

2+Chapter+2+Student+Version - MS&E 220 Probabilistic...

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1 MS&E 220 Probabilistic Analysis Lecture 2 Counting Samuel S. Chiu Stanford University Chapter Concepts Multiplication Principle: counting the combination of events Permutation Combination Conditional Reasoning Sequential Counting Partitioning, Scenario Analysis Dynamic Equation and the Pascal Triangle Counting the Complement: from which to deduce the original event Data Tree: organizing a multi-attribute data set Counting • Why? How well can you count?
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2 Counting Greeks used Astragali • Non-mathematical 960 A.D., Bishop Wibold enumerated 56 different outcomes of tossing three dices (without permutations). More later. If you consider ordering: 216 ways; appeared in a poem De Vetula in 13th century Counting Roman empire used lotteries to raise $, similar in the Middle Ages and Renaissance Private lotteries illegal because of cheating Later authorized to assist charities and the fine arts 16th and 17th century: mathematical analyses in dice, ball/urns, table games and lotteries Counting Mathematical analyses of card games not until the 18th century All the “games” involve equally likely outcomes, thus counting. – Chance of getting heads in a coin flip – Chance of getting an even number when tossing a six-sided die
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3 Simple concepts Multiplication rule and trees Permutation and its variants • Combinations Sequential reasoning Partition: divide and conquer Sequential counting: Dynamic Equations, it is not your grandfather’s Oldsmobile Balls and urns Data trees Multiplication rule: How many paths are there from root to end? a1 1 a1 2 a2 1 • • ak 1 a1 j a2 2 • • ak 2 a2 n 2 ak n k a2 1 a1 n 1 a2 2 a2 n 2 California License Plates 2GXT120: any ___ ___ ___ ___ ___ ___ ___ • 2SUE650: want SUE in it ___ ___ ___ ___ ___ ___ ___
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4 California License Plates Another way to visualize: use conceptual (virtual) bins Janet wants to have at least one J in her license plate. __ __ __ __ __ __ __ at least one J somewhere here 10 10 10 10 Understand the #s and Tell me what you learn Focusing on the alphabets only (three of them) Scenarios: Each characterizes some aspect(s) of possible "spellings" with three letters The focus is on the # of j 's # of j 's (exact) 0 1 2 3 How many are possible (in each component) How many? All possible "spellings" At least one j No j 's What did you learn? Mutually exclusive components • Partitioning Leading to: Examining the complement, it may be easier The complement of “at least one j ” is ______ # of j 's 0 1 2 3 How many are possible 15,625 1,875 75 1 All possible "spellings" x x x x Spellings with at least one j x x x Spellings with no j x
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5 Purple is special 4 color coins in a bag: B, G, Y, P Select one, flip it once – if it is not purple, return it to the bag – if it is purple, keep it for all subsequent flips Do it k times How many distinct color sequences are possible?
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