1 Stopping Times - Columbia University

1. (First passage/hitting times/Gambler’s ruin problem:) Suppose that X has a discrete state space and let ibe a xed state. Let ˝= minfn 0 : X n= ig: This is called the rst passage time of the process into state i. Also called the hitting time of the process to state i. More generally we can let Abe a collection of states such

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Introduction to Probability Models - University of North ...

which can be used to model the fortunes of an investor (or gambler) who always invests 1 and then receives a nonnegative integral return. Section 4.2 has additional material on Markov chains that shows how to modify a given chain when trying to determine such things as the probability that the chain ever enters a given class of

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Random Walk: A Modern Introduction - University of Chicago

5.1 Gambler’s ruin estimate 103 5.1.1 General case 106 5.2 One-dimensional killed walks 112 5.3 Hitting a half-line 115 6 Potential Theory 119 6.1 Introduction 119 6.2 Dirichlet problem 121 6.3 Difference estimates and Harnack inequality 125 6.4 Further estimates 132 6.5 Capacity, transient case 136 6.6 Capacity in two dimensions 144

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Practice with Expected Value and Fair Games Answers

2. The world famous gambler from Philadelphia, Señor Rick, proposes the following game of chance. You roll a fair die. If you roll a 1, then Señor Rick pays you $25. If you roll a 2, Señor Rick pays you $5. If you roll a 3, you win nothing. If you roll a 4 or a 5, you must pay Señor Rick $10, and if you roll a 6, you must pay Señor Rick $15.

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Chapter 4: Generating Functions - Auckland

sequence of repeating steps: for example, the Gambler’s Ruin from Section 2.7. The name probability generating function also gives us another clue to the role of the PGF. The PGF can be used to generate all the probabilities of the distribution. This is generally tedious and is not often an efficient way of calculating probabilities.

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Monte Carlo Method: Probability - Department of Scientific ...

Historically, the birth of probability arose when a gambler wrote to Fermat, asking him whether it was possible to settle up the bets in a particular dice game that had gotten interrupted. Fermat had some ideas, but he wrote to Pascal and between them they worked out methods that are still in use today. Burkardt Monte Carlo Method: Probability

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The 12 cognitive biases that prevent you from being rational

Gambler's Fallacy . It's called a fallacy, but it's more a glitch in our thinking. We tend to put a tremendous amount of weight on previous events, believing that they'll somehow influence future outcomes. The classic example is coin-tossing. After flipping heads, say, five consecutive times, our inclination is to predict an increase in

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Markov Chains - University of Cambridge

Markov Chains These notes contain material prepared by colleagues who have also presented this course at Cambridge, especially James Norris. The material mainly comes from books of