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I. Randomness

I. Randomness. Random (def.): Proceeding, made, or occurring without definite aim, reason or pattern. (OED) Randomness can have structure and, if so, we can use that structure to make predictions about subsequent outcomes.

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I. Randomness

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  1. I. Randomness • Random (def.): Proceeding, made, or occurring without definite aim, reason or pattern. (OED) • Randomness can have structure and, if so, we can use that structure to make predictions about subsequent outcomes. • In Statistics the outcomes of a random process follow a probability distribution (such that the probability of each outcome can be calculated), rather than a deterministic pattern. • Other words for random: stochastic, probabilitic. Seattle University - EMBA - Spring 2011

  2. II. Simulation (Monte Carlo Method) Idea: Build models to simulate random processes, and so investigate how the process “works”, i.e., given a set of random inputs with known distributions, determine empirically the distribution of output variables of interest. Seattle University - EMBA - Spring 2011

  3. II. Simulation (Monte Carlo Method) • Application examples: • the baby girl puzzle • the cereal box gimmick • multiple choice: luck, skill, or too good to be true? • the world series Seattle University - EMBA - Spring 2011

  4. III. Introduction to Probability • Some definitions: • trial: random phenomenon (process) happens • outcome: observed value(s) in a given trial • event: a combination of outcomes • sample space: the collection of all possible outcomes • relative frequency of an event: pr (A) = outcomes in which A occurs / total outcomes • independent trials: (informal) outcome of one trial doesn’t affect the outcome of another • disjoint events: events that have no outcomes in common Seattle University - EMBA - Spring 2011

  5. III. Introduction to Probability • Some definitions: • independent trials: (more formal)If knowledge that A has occurred affects our estimate of the probability that B will occur, we say that A and B are dependent.Otherwise we say that A and B are independent. Seattle University - EMBA - Spring 2011

  6. III. Introduction to Probability A key principle: Law of Large Numbers In repeated, independent trials with the same probability p of success in each trial, the chance that the percentage of successes differs from p by more than a fixed positive amount, ε > 0, converges to zero as the number of trials N goes to infinity, for every positive ε. Seattle University - EMBA - Spring 2011

  7. III. Introduction to Probability To think about: Example 1: Proportion odd numbers rolling a die. Example 2: Probability of boy-girl sequences A family has four children. Which sequence of boy and girl births is most likely: bbbb, bgbg, or gggg? Example 3: Length of world series. The length of the series must be between 4, 5, 6 or 7 games. What is the probability of each of these lengths? (Assuming that…) Seattle University - EMBA - Spring 2011

  8. .125 .250 .312 .313 III. Introduction to Probability Example 3: Length of world series. Simulation and theoretical results (assuming independence): Seattle University - EMBA - Spring 2011

  9. .125 .250 .312 .313 III. Introduction to Probability Example 3: Length of world series. Actual counts: Seattle University - EMBA - Spring 2011

  10. III. Introduction to Probability • Three concepts of probability: • Empirical probability ← relative frequency, justified by the LLN • Theoretical probability • Personal (or subjective) probability ← degree of belief Seattle University - EMBA - Spring 2011

  11. III. Introduction to Probability • Fundamental rules (or “axioms”) of probability: • P(A) is between 0 and 1 • P(sample space) = 1 : something must happen • P(A) = 1 – P(AC) : something either happens or doesn’t • If A and B are disjoint, then P(A or B) = P(A) + P(B) • If A and B are independent, then P(A & B) = P(A) P(B) Seattle University - EMBA - Spring 2011

  12. IV. Probability Rules General addition rules: P(A or B) = P(A) + P(B) – P(A & B) P(A xor B) = P(A) + P(B) – 2 [P(A & B)] Seattle University - EMBA - Spring 2011

  13. IV. Probability Rules Conditional probability: (a definition) P(B | A) ← probability of B given that A has occurred Conditional probability: (calculation) P(B | A) = P(A & B) / P(A) General multiplication rule: P(A & B) = P(A) P(B | A) or P(A & B) = P(B) P(A | B) Seattle University - EMBA - Spring 2011

  14. IV. Probability Rules Independence revisited: P(B | A) = P(B)  A and B are independentor P(A | B) = P(A)  A and B are independent Note: From the general multiplication rule, if A and B are independent, then P(A & B) = P(A) P(B) Seattle University - EMBA - Spring 2011

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