Statistics for Managers 5th Edition Chapter 4 Basic Probability 1

Chapter Topics Basic probability concepts Conditional probability Sample spaces and events, simple probability, joint probability Conditional 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 Joint 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 Tree Diagrams Black 2 24 26 Ace Not Ace Total Black 2 24 26 Red 2 24 26 Total 4 48 52 Ace Red Cards Not an Ace Full Deck of Cards Ace Black Cards 6 Not an Ace

 Special Events Impossible event Complement of event 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 Total Ace Red 2 24 26 Not an Ace Total Ace Red 2 24 26 Black 2 24 26 Total 4 48 52 Sample Space 8

Tree Diagram Event Possibilities Ace Red Cards Not an Ace Full Deck of Cards Ace Black Cards Not an Ace 9

Probability Certain Probability is the numerical measure of the likelihood that an event will occur Value is between 0 and 1 Sum of the probabilities of all mutually exclusive and collective exhaustive events is 1 1 .5 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: Each of the outcomes in the sample space is equally likely to occur number of event outcomes P(E)= total number of possible outcomes in sample space n(E) n(S) = e.g. P( ) = 2/36 (There are 2 ways to get one 6 and the other 4) 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 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)

Rules Of Multiplication 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 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 = 0.01 What is the probability that Bill filled a prescription that contained a mistake? 17

Bayes’s Theorem Adding up the parts of A in all the B’s Same Event 18

Bayes Theorem (cont.) P(B1 A) = (.20) (.10) (.20) (.10) + (.80) (.01) .02 .028 .71 71% 19

Bayes Theorem (cont.) Bill fills prescription Mary fills prescription (Prior) Bi .20 . 80 1.00 (Conditional) .10 .01 A Bi (Joint) A  Bi .020 .008 P(A) =.028 (Posterior) Bayes .02/.028=.71 .008/.028=.29 1.00 20

Chapter Summary Discussed basic probability concepts Sample spaces and events, simple probability, and joint probability Defined conditional probability Statistical independence, marginal probability Discussed Bayes’s theorem 21