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Business Statistics: A First Course 4th Edition Chapter 4 Basic Probability 1 Chapter Topics Basic probability concepts Conditional probability Sample spaces and events, simple probability, joint probability Statistical independence, marginal probability Bayes’s Theorem 2 Terminology Experiment- Process of Observation Outcome-Result of an Experiment Sample Space- All Possible Outcomes of a Given Experiment Event- A Subset of a Sample Space 3 Sample Spaces Collection of all possible outcomes e.g.: All six faces of a die: e.g.: All 52 cards in a deck: 4 Events Simple event Outcome from a sample space with one characteristic e.g.: A red card from a deck of cards Joint event Involves two outcomes simultaneously e.g.: An ace that is also red from a deck of cards 5 Visualizing Events Contingency Tables Ace Total Black Red 2 2 24 24 26 26 Total 4 48 52 Tree Diagrams Full Deck of Cards Not Ace Ace Red Cards Black Cards Not an Ace Ace Not an Ace 6 Special Events Null Event Impossible event e.g.: Club & diamond on one card draw Complement of event For event A, all events not in A Denoted as A’ e.g.: A: queen of diamonds A’: all cards in a deck that are not queen of diamonds 7 Contingency Table A Deck of 52 Cards Red Ace Ace Not an Ace Total Red 2 24 26 Black 2 24 26 Total 4 48 52 Sample Space 8 Tree Diagram Event Possibilities Full Deck of Cards Red Cards Ace Not an Ace Ace Black Cards Not an Ace 9 Probability Certain Probability is the numerical 1 measure of the likelihood that an event will occur Value is between 0 and 1 .5 Sum of the probabilities of all mutually exclusive and collective exhaustive events is 1 0 Impossible 10 Types of Probability •Classical (a priori) Probability P (Jack) = 4/52 •Empirical (Relative Frequency) Probability Probability it will rain today = 60% •Subjective Probability Probability that new product will be successful 11 Computing Probabilities The probability of an event E: number of event outcomes P(E)= total number of possible outcomes in sample space n(E) = e.g. P( ) = 2/36 n(S) (There are 2 ways to get one 6 and the other 4) Each of the outcomes in the sample space is equally likely to occur 12 Probability Rules 1 0 ≤ P(E) ≤ 1 Probability of any event must be between 0 and1 2 P(S) = 1 ; P(Ǿ) = 0 Probability that an event in the sample space will occur is 1; the probability that an event that is not in the sample space will occur is 0 3 P (E) = 1 – P(E) Probability that event E will not occur is 1 minus the probability that it will occur 13 Rules of Addition 4 5. Special Rule of Addition P (AuB) = P(A) + P(B) if and only if A and B are mutually exclusive events General Rule of Addition P (AuB) = P(A) + P(B) – P(AnB) 14 Rules Of Multiplication 6 7 Special Rule of Multiplication P (AnB) = P(A) x P(B) if and only if A and B are statistically independent events General Rule of Multiplication P (AnB) = P(A) x P(B/A) 15 Conditional Probability Rule Conditional Probability Rule P(B/A) = P (AnB)/ P(A) This is a rewrite of the formula for the general rule of multiplication. 16 Bayes Theorem P(B1) = probability that Bill fills prescription = .20 P(B2) = probability that Mary fills prescription = .80 P(A B1) = probability mistake Bill fills prescription = 0.10 P(A B2) = probability mistake Mary fills prescription = 0.0 What is the probability that Bill filled a prescription that contained a mistake? 17 Bayes’s Theorem P A | Bi P Bi P Bi | A = P A | B1 P B1 P A | Bk P Bk P Bi and A = P A Adding up Same Event the parts of A in all the B’s 18 Bayes Theorem (cont.) (.20) (.10) P(B1 A) = (.20) (.10) + (.80) (.01) .02 = = .71 = 71% .028 19 Bayes Theorem (cont.) (Prior) Bi Bill fills prescription (Conditional) A Bi (Joint) A Bi .20 .10 .020 Mary fills prescription. 80 .01 .008 1.00 P(A) =.028 (Posterior) Bayes .02/.028=.71 .008/.028=.29 1.00 20 Chapter Summary Discussed basic probability concepts Defined conditional probability Sample spaces and events, simple probability, and joint probability Statistical independence, marginal probability Discussed Bayes’s theorem 21