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1 Probability Models De…nition 1 A probability model is a mathematical description of an experiment consisting of two parts: a disjoint listing of all outcomes of the experiment and an assignment of probabilities to each outcome. A probability model is similar to a sample space but can describe more complex outcomes because there is no longer the requirement that outcomes be equally likely. Recall that the sample space for rolling a pair of dice has 36 outcomes. (1; 1) (2; 1) (3; 1) (4; 1) (5; 1) (6; 1) (1; 2) (2; 2) (3; 2) (4; 2) (5; 2) (6; 2) (1; 3) (2; 3) (3; 3) (4; 3) (5; 3) (6; 3) (1; 4) (2; 4) (3; 4) (4; 4) (5; 4) (6; 4) (1; 5) (2; 5) (3; 5) (4; 5) (5; 5) (6; 5) (1; 6) (2; 6) (3; 6) (4; 6) (5; 6) (6; 6) Example 1 Construct the probability model for the sum of two dice. sum x 2 3 4 5 6 7 1 2 3 4 5 6 P (x) 36 36 36 36 36 36 sum x 8 9 10 11 12 5 4 3 2 1 P (x) 36 36 36 36 36 Exercise 1 Construct the sample space for rolling one 3-sided die and one 4sided die. Construct the probability model for the sum of the two dice. 2 Random Variables De…nition 2 A random variable assumes a value based on the outcome of a random event. There are two types of random variables. Each type creates a very di¤erent probability model with its own set of rules and computations. De…nition 3 Discrete random variables can take one of a …nite number of distinct outcomes. Discrete random variables jump from one state to the next with nothing in between. Example 2 The number of innings in a baseball game, number of players on a basketball team and the number of cards in a deck are all discrete random variables. De…nition 4 Continuous random variables can take any numeric value within a range of values. Example 3 The weight of a baseball (in ounces), height of a player and the time it takes (in seconds) to run 40 yards are all examples of continuous random variables. 1 Exercise 2 Consider an experiment whose population is the set of all Kennesaw State University students. Which of these variables are discrete and which are continuous? number of classes the student is enrolled in this semester; height in inches; annual income in dollars; GPA; number of miles driven to campus. Discrete probability models can use sample spaces to compute probabilities. That does not work with continuous random variables which have an in…nite number of values. The normal curve is an example of a continuous probability model. In a continuous model, probability is computed as area under the curve. As a general rule, such computations require calculus! Fortunately, the normal curve is so important and well known there are many tools to compute probabilities without resorting to calculus. In fact, that is what we were doing every time we used Excel to perform computations in the normal curve. 3 Exercises 1. Moore text: page 179: 9.26, 9.30, 9.34, 9.35, 9.38 2. For Kevin Garnett’s career stats (up to 2011), determine if each random variable is continuous or discrete. 2 games played; games started; minutes played; …eld goals; …eld goals attempted; …eld goal percentage; 3-point …eld goals; 3-point …eld goals attempted; 3-point …eld goal percentage; free throws; free throw attempts; free throw percentage; o¤ensive rebounds; defensive rebounds; total rebounds; assists; steals; blocks; turnovers; personal fouls; blocks; total points. 3