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Chapter 4: Basic Probability • Chapter Goal: – Explain basic probability concepts and definitions – Use contingency tables to view a sample space – Apply common rules of probability – Compute conditional probabilities – Determine whether events are statistically independent Definitions: • Probability – the chance or likelihood that an uncertain (particular) event will occur • Probability is always between 0 and 1, inclusive • There are three approaches to assessing the probability of un uncertain event: 1. a priori classical probability– based of a prior knowledge prob. of occurrence X number of occurance of the event T total number of possible outcomes 2. empirical classical probability– based on observed data prob. of occurrence number of favorable outcomes observed total number of outcomes observed 3. subjective probability -- an individual judgment or opinion about the probability of occurrence Basic Concepts: • Sample Space – the collection of all possible events • An Event – Each possible type of occurrence or outcome from the sample space • Simple Event – an event that can be described by a single characteristic • Complement of an event A -- All outcomes that are not part of event A • Joint event --Involves events that can be described by two or more characteristics simultaneously Basic Concepts: Continued • Mutually exclusive events: Events that cannot occur together • Collectively exhaustive events – One of the events must occur – The set of events covers the entire sample space • The probability of any event must be between 0 and 1, inclusively. That is: 0 ≤ P(A) ≤ 1 for any event A • The sum of the probabilities of all mutually exclusive and collectively exhaustive events is 1 exhaustive. That is, if A, B, and C are mutually exclusive and collectively exhaustive event, then (the entire sample space) P(A) P(B) P(C) 1 Contingency Tables: • A sample space can be presented by a C.T. • It is very useful for the study of empirical probabilities • Example: Let’s say 400 managers were surveyed about booking airline tickets and researching prices of tickets in the internet Booked Airline Ticket in the Internet Sample Space: Total Number of Managers Surveyed Yes, B1 Researched Prices in the Internet No, B2 Totals Yes, A1 88 124 212 No, A2 20 168 188 Totals 108 292 400 • Computing Simple (Marginal) Probobilities. P(A) P(A and B1) P(A and B2 ) P(A and Bk ) • Where B1, B2, …, Bk are k mutually exclusive and collectively exhaustive events • Probability of a joint event, A and B: P( A and B) number of outcomes satisfying A and B total number of possible outcomes • How many simple events are in my example? • How many joint events are in my example? Rules of Probability: 1. General Addition Rule P(A or B) = P(A) + P(B) - P(A and B) If A and B are mutually exclusive, then P(A and B) = 0, so the rule can be simplified P(A or B) = P(A) + P(B) If A and B are collectively exhaustive, then P(A or B) = P(A) + P(B)=1 2. A conditional probability is the probability of one event, given that another event has occurred: P(A and B) P(A | B) P(B) P(A and B) P(B | A) P(A) The conditional probability of A given that B has occurred The conditional probability of B given that A has occurred Where P(A and B) = joint probability of A and B P(A) = marginal probability of A P(B) = marginal probability of B 3. Multiplication Rule– for two events A and B P(A and B) P(A | B) P(B) 4. Two events A and B are statistically independent if the probability of one event is unchanged by the knowledge that other even occurred. That is: P(A | B) P(A) or P(B | A) P(B) 5. Then the multiplication rule for two statistically independent events is: P(A and B) P(A) P(B) • Let look at problem 4.26, page 148