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Introduction to Probability BUSA 2100, Sections 4.0, 4.1, 4.2 Need For and Uses Of Probability Probabilities are necessary because we live in an uncertain world. Probabilities are a way of quantifying uncertainty. Definition: A probability is a numerical measure of the likelihood or chance that an event will occur. Probability was first used in the context of gambling, i.e. cards, coins, and dice. More Uses of Probability Cards, coins, and dice still provide good examples for explaining probabilities. Other uses of probability include: weather forecasting, biology, political science, insurance, investments, & sales. Life and car insurance rates are based upon life expectancies and probabilities of auto accidents. Events and Probabilities Definition: An event is one or more of the possible outcomes of an activity, e.g. “even number on a die”. Notation: P(E) represents the probability that event E will occur.” The probability of an event is always between 0 and 1. (fraction or decimal) Types of Events Def.: The complement of an event E, denoted by EC, is the opposite of event E. Example: If E = “will rain today”, then EC = “will not rain today.” Formula for EC? Definition: Two or more events are mutually exclusive if only one of them can occur at a time. Types of Events, Page 2 Example 1: F = “car made by Ford Motor Co.”, G = “car made by General Motors” are mutually exclusive events. Example 2: D = “person who has a daughter”, “S = “person who has a sister” are not mutually exclusive events. Definition: A set of events is exhaustive if it includes all possible outcomes. Types of Events, Page 3 Example 1: For primary colors, R = “red”, B = “blue”, Y = “yellow” are exhaustive events. Example 2: S = “sophomore”, J = “junior” are not exhaustive events. If a set of events is mutually exclusive and exhaustive, the probabilities of these events must sum to 1. Types of Events, Page 4 Definition: Two or more events are equally likely if each event has the same probability of occurrence. Examples: A 1, 2, 3, or 4 on a die; a boy or girl baby. Ways to Obtain Probabilities First method: The classical formula, P(E) = (number of outcomes pertaining to event E) / (total number of possible outcomes). The classical formula is true only if the outcomes are mutually exclusive, exhaustive, and equally likely. Obtaining Probabilities, p. 2 Example 1: If 2 dice are rolled, what is the probability that the sum of the numbers on the dice will be 8? Obtaining Probabilities, p. 3 Example 2: In a family of 3 children, what is the probability of 2 boys &1 girl? Obtaining Probabilities, p. 4 Advantage: Classical formula has nearly perfect accuracy. Disadvantage: Often a list or count of all possible outcomes is not practical. Obtaining Probabilities, p. 5 Second method: The relative frequency method -- using relative frequencies of past occurrences as probabilities for the present and future. Advantages: Method usually has very good accuracy, is easy to use, and is applicable to a wide variety of situations. Obtaining Probabilities, p. 6 Example. Past daily TV sales for a firm: Daily Sales No. of Days (Frequencies) 50 9 55 18 60 36 65 27 Obtaining Probabilities, p. 7 Third method: Subjective method -- a probability based on relevant information, experience, judgment, and intuition, but not based on a specific formula. It is an informed estimate. Obtaining Probabilities, p. 8 Example 1: What is the probability that the inflation rate will be less than 4% next year? Obtaining Probabilities, p. 9 Example 2: What is the probability that the Braves will the NL championship? The subjective method is the least accurate method of obtaining probabilities. But subjective probabilities are better than none at all.