Chapter Chapter Goals. Assessing Probability. Important Terms. Events. Sample Space. Chapter 4 Basic Probability

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1 Chapter 4 4- Chapter Goals Chapter 4 Basic Probability fter completing this chapter, you should be able to: Explain basic probability concepts and definitions Use contingency tables to view a sample space pply common rules of probability Determine whether events are statistically independent Compute conditional probabilities Use Bayes Theorem for conditional probabilities Yandell GSB 50 Chap 4- Yandell GSB 50 Chap 4- Important Terms ssessing Probability Probability the chance that an uncertain event will occur (always between 0 and ) Experiment a process of obtaining outcomes for uncertain events Elementary Event the most basic outcome possible from a simple experiment Sample Space the collection of all possible elementary outcomes There are three approaches to assessing the probability of un uncertain event: a priori classical probability X number of ways the event can occur probabilit y of occurrence = = T total number of elementary outcomes empirical classical probability number of favorable outcomes observed probabilit y of occurrence = total number of outcomes observed 3 subjective probability an individual judgment or opinion about the probability of occurrence Yandell GSB 50 Chap 4-3 Yandell GSB 50 Chap 4-4 Sample Space The Sample Space is the collection of all possible outcomes eg ll 6 faces of a die: eg ll 5 cards of a bridge deck: Simple event Events n outcome from a sample space with one characteristic eg, red card from a deck of cards Complement of an event (denoted ) ll outcomes that are not part of event eg, ll cards that are not diamonds Joint event Involves two or more characteristics simultaneously eg, n ace that is also red from a deck of cards Yandell GSB 50 Chap 4-5 Yandell GSB 50 Chap 4-6

2 Chapter 4 4- Sample Space Contingency Tables Tree Diagrams Full Deck of 5 Cards Visualizing Events ce Not ce Total Black 4 6 Red 4 6 Total Black Card Red Card ce Not an ce ce Not an ce 4 4 Sample Space Yandell GSB 50 Chap 4-7 Elementary Events automobile consultant records fuel type and vehicle type for a sample of vehicles Fuel types: Gasoline, Diesel 3 Vehicle types: Truck, Car, SUV 6 possible elementary events: e Gasoline, Truck e Gasoline, Car e 3 Gasoline, SUV e 4 Diesel, Truck e 5 Diesel, Car Diesel, SUV e 6 Gasoline Diesel Truck Yandell GSB 50 Chap 4-8 SUV Car Truck SUV Car e e e 3 e 4 e 5 e 6 Probability Concepts Mutually Exclusive Events If occurs, then E cannot occur and E have no common elements E Black Cards Red Cards card cannot be Black and Red at the same time Yandell GSB 50 Chap 4-9 Collectively Exhaustive Events Collectively exhaustive events One of the events must occur The set of events covers the entire sample space example: = aces; B = black cards; C = diamonds; D = hearts Events, B, C and D are collectively exhaustive (but not mutually exclusive an ace may also be a heart) Events B, C and D are collectively exhaustive and also mutually exclusive Yandell GSB 50 Chap 4-0 Probability Concepts Independent and Dependent Events Independent: Occurrence of one does not influence the probability of occurrence of the other Dependent: Occurrence of one affects the probability of the other Independent vs Dependent Events Independent Events = heads on one flip of fair coin E = heads on second flip of same coin Result of second flip does not depend on the result of the first flip Dependent Events = rain forecasted on the news E = take umbrella to work Probability of the second event is affected by the occurrence of the first event Yandell GSB 50 Chap 4- Yandell GSB 50 Chap 4-

3 Chapter Individual Values Rules of Probability Rules for Individual Values and Sum Sum of ll Values 0 P(e i ) P(e i ) = For any event e i Yandell GSB 50 Chap 4-3 T i= where: T = Number of elementary events in the sample space e i = i th elementary event ddition Rule for Elementary Events The probability of an event E is equal to the sum of the probabilities of the elementary events forming E That is, if: then: E = {e, e, e 3 } P(E) = P(e ) + P(e ) + P(e 3 ) Yandell GSB 50 Chap 4-4 Complement Rule ddition Rule for Two Events The complement of an event E is the collection of all possible elementary events not contained in event E The complement of event E is represented by E Complement Rule: P( E) P(E) = E Or, E P(E) + P(E) = ddition Rule: P( or B) = P() + P(B) - P( and B) + B = P( or B) = P() + P(B) - P( and B) Don t count common elements twice! B Yandell GSB 50 Chap 4-5 Yandell GSB 50 Chap 4-6 ddition Rule Example P(Red or ce) = P(Red) +P(ce) - P(Red and ce) ddition Rule for Mutually Exclusive Events If and B are mutually exclusive, then = 6/5 + 4/5 - /5 = 8/5 Color Type Red Black Total ce 4 Non-ce Total Don t count the two red aces twice! So P( and B) = 0 P( or B) = P() + P(B) - P( and B) = P() + P(B) = 0 B if mutually exclusive Yandell GSB 50 Chap 4-7 Yandell GSB 50 Chap 4-8

4 Chapter Conditional Probability Conditional probability for any two events, B: P( andb) P( B) = P(B) where P(B) > 0 Of the cars on a used car lot, 70% have air conditioning (C) and 40% have a CD player (CD) 0% of the cars have both What is the probability that a car has a CD player, given that it has C? ie, we want to find P(CD C) Yandell GSB 50 Chap 4-9 Yandell GSB 50 Chap 4-0 Of the cars on a used car lot, 70% have air conditioning (C) and 40% have a CD player (CD) 0% of the cars have both CD No CD Total C 5 7 No C 3 Total P(CDand C) P(CD C) = = = 857 P(C) 7 Given C, we only consider the top row (70% of the cars) Of these, 0% have a CD player 0% of 70% is about 857% CD No CD Total C 5 7 No C 3 Total P(CDand C) P(CD C) = = = 857 P(C) 7 Yandell GSB 50 Chap 4- Yandell GSB 50 Chap 4- For Independent Events: Conditional probability for independent events, B: P( B) = P() where P(B) > 0 P(B ) = P(B) where P() > 0 Multiplication Rules Multiplication rule for two events and B: P( and B) = P()P(B ) Note: If and B are independent, then and the multiplication rule simplifies to P( and B) = P()P(B) P(B ) = P(B) Yandell GSB 50 Chap 4-3 Yandell GSB 50 Chap 4-4

5 Chapter Tree Diagram Example Bayes Theorem Gasoline P( ) = 08 Truck: P(E 3 ) = 0 Car: P(E 4 ) = 05 SUV: P(E 5 ) = 03 P( and E 3 ) = 08 x 0 = 06 P( and E 4 ) = 08 x 05 = 040 P( and E 5 ) = 08 x 03 = 04 P(E)P(B E i i) P(E i B) = P(E )P(B E ) + P(E )P(B E ) + K+ P(E )P(B E ) k k Diesel P(E ) = 0 Truck: P(E 3 E ) = 06 Car: P(E 4 E ) = 0 SUV: P(E 5 E ) = 03 P(E and E 3 ) = 0 x 06 = 0 P(E and E 4 ) = 0 x 0 = 00 P(E 3 and E 4 ) = 0 x 03 = 006 where: E i = i th event of interest of the T possible events B = new event that might impact P(E i ) Events to E k are mutually exclusive and collectively exhaustive Yandell GSB 50 Chap 4-5 Yandell GSB 50 Chap 4-6 drilling company has estimated a 40% chance of striking oil for their new well detailed test has been scheduled for more information Historically, 60% of successful wells have had detailed tests, and 0% of unsuccessful wells have had detailed tests Given that this well has been scheduled for a detailed test, what is the probability that the well will be successful? Yandell GSB 50 Chap 4-7 Let S = successful well and U = unsuccessful well P(S) = 4, P(U) = 6 (prior probabilities) Define the detailed test event as D Conditional probabilities: P(D S) = 6 P(D U) = Revised probabilities Event S (successful) U (unsuccessful) Prior 4 6 Conditional 6 Joint 4*6 = 4 6* = Sum = 36 Revised 4/36 = 67 /36 = 33 Yandell GSB 50 Chap 4-8 Given the detailed test, the revised probability of a successful well has risen to 67 from the original estimate of 4 Event S (successful) U (unsuccessful) Prior 4 6 Conditional 6 Joint 4*6 = 4 6* = Sum = 36 Revised 4/36 = 67 /36 = 33 Yandell GSB 50 Chap 4-9 Chapter Summary Discussed basic probability concepts Sample spaces and events, contingency tables, simple probability, and joint probability Examined basic probability rules General addition rule, addition rule for mutually exclusive events, rule for collectively exhaustive events Defined conditional probability Statistical independence, marginal probability, decision trees, and the multiplication rule Yandell GSB 50 Chap 4-30