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MODULE 1 THE WHOLE MODULE

MODULE 1 THE WHOLE MODULE - 1 Probability Basics Dave...

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1. Probability Basics Dave Goldsman Georgia Institute of Technology, Atlanta, GA, USA 10/19/10 Goldsman 10/19/10 1 / 98
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Outline 1 Intro / Examples 2 Set Theory 3 Experiments 4 Probability 5 Finite Sample Spaces 6 Counting Techniques 7 Counting Applications 8 Conditional Probability and Independence 9 Bayes Theorem Goldsman 10/19/10 2 / 98
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Intro / Examples Mathematical Models for describing observable phenomena: Deterministic Probabilistic Deterministic Models Ohm’s Law ( I = E/R ) Drop an object from height h 0 . After t sec, height is h ( t ) = h 0 - 16 t 2 . Deposit $1000 in a continuously compounding checking 3% account. At time t , it’s worth $1000 e . 03 t . Goldsman 10/19/10 3 / 98
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Intro / Examples Probabilistic Models — Involve uncertainty How much snow will fall tomorrow? Will IBM make a profit this year? Should I buy a call or put option? Can I win in blackjack if I use a certain strategy? What is the cost-effectiveness of a new drug? Which horse will win the Kentucky Derby? Goldsman 10/19/10 4 / 98
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Intro / Examples Some Cool Examples 1. Birthday Problem — Assume all 365 days have equal probability of being a person’s birthday (ignore Feb 29). Then. . . If there are 23 people in the room, the odds are better than 50–50 that there will be a match. If there are 50 people, the probability is about 97%! 2. Monopoly — In the long run, the property having the highest probability of being landed on is Illinois Ave. Goldsman 10/19/10 5 / 98
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Intro / Examples 3. Poker — Pick 5 cards from a standard deck. Then P ( exactly 2 pairs ) 0 . 0475 P ( full house ) 0 . 00144 P ( flush ) 0 . 00198 4. Stock Market — Monkeys randomly selecting stocks could have outperformed most market analysts during the past year. Goldsman 10/19/10 6 / 98
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Intro / Examples 5. A couple has two kids and at least one is a boy. What’s the probability that BOTH are boys? Possibilities: GG, BG, GB, BB. Eliminate GG since we know that there’s at least one boy. Then P ( BB ) = 1 / 3 . 6. Vietnam Lottery 7. Ask Marilyn. You are a contestant at a game show. Behind one of three doors is a car; behind the other two are goats. You pick door A. Monty Hall opens door B and reveals a goat. Monty offers you a chance to switch to door C. What should you do? Goldsman 10/19/10 7 / 98
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Intro / Examples Working Definitions Probability — Methodology that describes the random variation in systems. (We’ll spend about 40% of our time on this.) Statistics — Uses sample data to draw general conclusions about the population from which the sample was taken. (60% of our time.) Goldsman 10/19/10 8 / 98
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Set Theory Outline 1 Intro / Examples 2 Set Theory 3 Experiments 4 Probability 5 Finite Sample Spaces 6 Counting Techniques 7 Counting Applications 8 Conditional Probability and Independence 9 Bayes Theorem Goldsman 10/19/10 9 / 98
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Set Theory The Joy of Sets Definition: A set is a collection of objects. Members of a set are called elements .
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