Introduction to Probability

1. Introduction to Probability BUSA 2100, Sections 4.0, 4.1, 4.2

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.

3. More Uses of Probability • Cards, coins, and dice still provide good examples for explaining probabilities. • Other uses of probability include: wea-ther forecasting, biology, political sci-ence, insurance, investments, & sales. • Life and car insurance rates are based upon life expectancies and probabilities of auto accidents.

4. 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)

5. Types of Events • Def.: The complement of an event E, denoted by EC, is theopposite 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.

6. 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 sis-ter” are not mutually exclusive events. • Definition: A set of events is exhaustive if it includes all possible outcomes.

7. 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” arenot exhaustive events. • If a set of events is mutually exclusive and exhaustive, the probabilities of these events must sum to 1.

8. 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.

9. 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.

10. 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?

11. Obtaining Probabilities, p. 3 • Example 2: In a family of 3 children, what is the probability of 2 boys &1 girl?

12. Obtaining Probabilities, p. 4 • Advantage: Classical formula has nearly perfect accuracy. • Disadvantage: Often a list or count of all possible outcomes is not practical.

13. Obtaining Probabilities, p. 5 • Second method: The relative fre-quencymethod -- 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.

14. 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

15. Obtaining Probabilities, p. 7 • Third method: Subjectivemethod -- a probability based on relevant information, experience, judgment, and intuition, but not based on a specific formula. • It is an informed estimate.

16. Obtaining Probabilities, p. 8 • Example 1: What is the probability that the inflation rate will be less than 4% next year?

17. 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.