Probability Theory and Game of Chance

Probability Theory and Game of Chance Jingjing Xu April 24, 2012 I. INTRODUCTION Probability theory is the mathematical foundation of statistics, and it can be applied to many areas requiring large data analysis. Curiously, that the study on probability theory has its root in parlor games and gambling. In 17th century, dice gambling was a very common entertainment among the upper class. An Italian mathematician and gambler Gerolamo Cardano founded the concept of probability by studying the rules of rolling dice: since a die is a cube with each of its six faces showing a different number from 1 to 6, when it is rolled, the probability of seeing each number is equal. Therefore, some of the gamblers began to wonder, that taking a pair of dice and rolling them a couple of times, which has the larger probability of seeing a sum of 9 or seeing a sum of 10? What about seeing double sixes? In a correspondence between Blaise Pascal and Pierre Fermat, the problems were resolved, and this triggered the first theorem in the modern theory of probability. II. BASIC DEFINITIONS Definition 1 In probability theory, the sample space, often denoted Ω, of an experiment is the set of all possible outcomes. Thus, for each element x∈Ω, a probability value f(x) attached, we have: 1.1. f(x)∈0, 1, ∀ x∈Ω; 1.2. x∈Ωfx=1. Which means, a probability value should be more than 0 and less than 1, and the sum of the probabilities over all values x in the sample space is equal to 1. A event is defined as any subset E of the sample space Ω, and the probability of E is written as P(E): 1.3. PE=x∈Efx Thus, the probability of the entire sample space is 1, and the probability of the null event is

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