1. Statistics for Managers 4th Edition Chapter 4 Basic Probability

2. Chapter Topics • Basic probability concepts • Sample spaces and events, simple probability, joint probability • Conditional probability • Statistical independence, marginal probability • Bayes’s Theorem

3. 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

4. Sample Spaces • Collection of all possible outcomes • e.g.: All six faces of a die: • e.g.: All 52 cards in a deck:

5. 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

6. Visualizing Events • Contingency Tables • Tree Diagrams 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 Not an Ace

7.   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

8. Contingency Table A Deck of 52 Cards Red Ace Not an Ace Total Ace Red 2 24 26 Black 2 24 26 Total 4 48 52 Sample Space

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

10. Probability Certain 1 • 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 .5 0 Impossible

11. 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

12. number of event outcomes P(E)= total number of possible outcomes in sample space n(E) = n(S) Computing Probabilities • The probability of an event E: • Each of the outcomes in the sample space is equally likely to occur e.g. P() = 2/36 (There are 2 ways to get one 6 and the other 4)

13. 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

14. Rules of Addition (Union of Events) • 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)

15. Rules Of Multiplication Intersection of Events) • 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)

16. 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.

17. 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 Bayes Theorem What is the probability that Bill filled a prescription that contained a mistake?

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

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

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

21. 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