1. 1 1 Slide © 2007 Thomson South-Western. All Rights Reserved Chapter 4 Introduction to Probability Experiments, Counting Rules, and Assigning Probabilities Events and Their Probability Some Basic Relationships of Probability Conditional Probability Bayes’ Theorem

2. 2 2 Slide © 2007 Thomson South-Western. All Rights Reserved Probability as a Numerical Measure of the Likelihood of Occurrence 0 1.5 Increasing Likelihood of Occurrence Probability: The event is very unlikely to occur. The event is very unlikely to occur. The occurrence of the event is just as likely as just as likely as it is unlikely. The occurrence of the event is just as likely as just as likely as it is unlikely. The event is almost certain to occur. The event is almost certain to occur.

3. 3 3 Slide © 2007 Thomson South-Western. All Rights Reserved An Experiment and Its Sample Space An experiment is any process that generates An experiment is any process that generates well-defined outcomes (e.g. throwing a dice). well-defined outcomes (e.g. throwing a dice). An experiment is any process that generates An experiment is any process that generates well-defined outcomes (e.g. throwing a dice). well-defined outcomes (e.g. throwing a dice). The sample space for an experiment is the set of The sample space for an experiment is the set of all experimental outcomes (in the experiment of throwing a dice, the sample space is {1,2,3,4,5,6} ). all experimental outcomes (in the experiment of throwing a dice, the sample space is {1,2,3,4,5,6} ). The sample space for an experiment is the set of The sample space for an experiment is the set of all experimental outcomes (in the experiment of throwing a dice, the sample space is {1,2,3,4,5,6} ). all experimental outcomes (in the experiment of throwing a dice, the sample space is {1,2,3,4,5,6} ). An experimental outcome is also called a sample An experimental outcome is also called a sample point (e.g. a sample point when you throw a dice point (e.g. a sample point when you throw a dice is to get a 5). An experimental outcome is also called a sample An experimental outcome is also called a sample point (e.g. a sample point when you throw a dice point (e.g. a sample point when you throw a dice is to get a 5).

4. 4 4 Slide © 2007 Thomson South-Western. All Rights Reserved A Counting Rule for Multiple-Step Experiments If an experiment consists of a sequence of k steps If an experiment consists of a sequence of k steps in which there are n 1 possible results for the first step, in which there are n 1 possible results for the first step, n 2 possible results for the second step, and so on, n 2 possible results for the second step, and so on, then the total number of experimental outcomes is then the total number of experimental outcomes is given by ( n 1 )( n 2 )... ( n k ). given by ( n 1 )( n 2 )... ( n k ). A helpful graphical representation of a multiple-step A helpful graphical representation of a multiple-step experiment is a tree diagram. experiment is a tree diagram.

5. 5 5 Slide © 2007 Thomson South-Western. All Rights Reserved Assigning Probabilities Classical Method Relative Frequency Method Subjective Method Assigning probabilities based on the assumption Assigning probabilities based on the assumption of equally likely outcomes of equally likely outcomes Assigning probabilities based on experimentation Assigning probabilities based on experimentation or historical data or historical data Assigning probabilities based on judgment Assigning probabilities based on judgment

6. 6 6 Slide © 2007 Thomson South-Western. All Rights Reserved Remember that… n The probability of any outcome in an experiment must be between 0 and 1 n The sum of the probabilities of all possible outcomes in an experiment equals 1.

7. 7 7 Slide © 2007 Thomson South-Western. All Rights Reserved Classical Method If an experiment has n possible outcomes, this method If an experiment has n possible outcomes, this method would assign a probability of 1/ n to each outcome. Experiment: Rolling a die Sample Space: S = {1, 2, 3, 4, 5, 6} Probabilities: Each sample point has a 1/6 chance of occurring 1/6 chance of occurring Example

8. 8 8 Slide © 2007 Thomson South-Western. All Rights Reserved Relative Frequency Method Number of Polishers Rented Number of Days 01234 4 61810 2 Lucas Tool Rental would like to assign probabilities to the number of car polishers it rents each day. Office records show the following frequencies of daily rentals for the last 40 days. Example: Lucas Tool Rental

9. 9 9 Slide © 2007 Thomson South-Western. All Rights Reserved Subjective Method When economic conditions and a company’s When economic conditions and a company’s circumstances change rapidly it might be circumstances change rapidly it might be inappropriate to assign probabilities based solely on inappropriate to assign probabilities based solely on historical data. historical data. We can use any data available as well as our We can use any data available as well as our experience and intuition, but ultimately a probability experience and intuition, but ultimately a probability value should express our degree of belief that the value should express our degree of belief that the experimental outcome will occur. experimental outcome will occur. The best probability estimates often are obtained by The best probability estimates often are obtained by combining the estimates from the classical or relative combining the estimates from the classical or relative frequency approach with the subjective estimate. frequency approach with the subjective estimate.

10. 10 Slide © 2007 Thomson South-Western. All Rights Reserved An event is a collection of sample points. An event is a collection of sample points. The probability of any event is equal to the sum of The probability of any event is equal to the sum of the probabilities of the sample points in the event. the probabilities of the sample points in the event. The probability of any event is equal to the sum of The probability of any event is equal to the sum of the probabilities of the sample points in the event. the probabilities of the sample points in the event. If we can identify all the sample points of an If we can identify all the sample points of an experiment and assign a probability to each, we experiment and assign a probability to each, we can compute the probability of an event. can compute the probability of an event. If we can identify all the sample points of an If we can identify all the sample points of an experiment and assign a probability to each, we experiment and assign a probability to each, we can compute the probability of an event. can compute the probability of an event. Events and Their Probabilities

11. 11 Slide © 2007 Thomson South-Western. All Rights Reserved Some Basic Relationships of Probability There are some basic probability relationships that can be used to compute the probability of an event without knowledge of all the sample point probabilities. Complement of an Event Complement of an Event Intersection of Two Events Intersection of Two Events Mutually Exclusive Events Mutually Exclusive Events Union of Two Events

12. 12 Slide © 2007 Thomson South-Western. All Rights Reserved The complement of A is denoted by A c. The complement of A is denoted by A c. The complement of event A is defined to be the event The complement of event A is defined to be the event consisting of all sample points that are not in A. consisting of all sample points that are not in A. The complement of event A is defined to be the event The complement of event A is defined to be the event consisting of all sample points that are not in A. consisting of all sample points that are not in A. Complement of an Event Event A AcAcAcAc Sample Space S Sample Space S VennDiagram

13. 13 Slide © 2007 Thomson South-Western. All Rights Reserved The union of events A and B is denoted by A  B  The union of events A and B is denoted by A  B  The union of events A and B is the event containing The union of events A and B is the event containing all sample points that are in A or B or both. all sample points that are in A or B or both. The union of events A and B is the event containing The union of events A and B is the event containing all sample points that are in A or B or both. all sample points that are in A or B or both. Union of Two Events Sample Space S Sample Space S Event A Event B

14. 14 Slide © 2007 Thomson South-Western. All Rights Reserved The intersection of events A and B is denoted by A  The intersection of events A and B is denoted by A  The intersection of events A and B is the set of all The intersection of events A and B is the set of all sample points that are in both A and B. sample points that are in both A and B. The intersection of events A and B is the set of all The intersection of events A and B is the set of all sample points that are in both A and B. sample points that are in both A and B. Sample Space S Sample Space S Event A Event B Intersection of Two Events Intersection of A and B

15. 15 Slide © 2007 Thomson South-Western. All Rights Reserved The addition law provides a way to compute the The addition law provides a way to compute the probability of event A, or B, or both A and B occurring. probability of event A, or B, or both A and B occurring. The addition law provides a way to compute the The addition law provides a way to compute the probability of event A, or B, or both A and B occurring. probability of event A, or B, or both A and B occurring. Addition Law The law is written as: The law is written as: P ( A  B ) = P ( A ) + P ( B )  P ( A  B 

16. 16 Slide © 2007 Thomson South-Western. All Rights Reserved Mutually Exclusive Events Two events are said to be mutually exclusive if the Two events are said to be mutually exclusive if the events have no sample points in common. events have no sample points in common. Two events are said to be mutually exclusive if the Two events are said to be mutually exclusive if the events have no sample points in common. events have no sample points in common. Two events are mutually exclusive if, when one event Two events are mutually exclusive if, when one event occurs, the other cannot occur. occurs, the other cannot occur. Two events are mutually exclusive if, when one event Two events are mutually exclusive if, when one event occurs, the other cannot occur. occurs, the other cannot occur. Sample Space S Sample Space S Event A Event B

17. 17 Slide © 2007 Thomson South-Western. All Rights Reserved Mutually Exclusive Events If events A and B are mutually exclusive, P ( A  B  = 0. If events A and B are mutually exclusive, P ( A  B  = 0. The addition law for mutually exclusive events is: The addition law for mutually exclusive events is: P ( A  B ) = P ( A ) + P ( B ) there’s no need to include “  P ( A  B  ” there’s no need to include “  P ( A  B  ”

18. 18 Slide © 2007 Thomson South-Western. All Rights Reserved The probability of an event given that another event The probability of an event given that another event has occurred is called a conditional probability or joint probability. has occurred is called a conditional probability or joint probability. The probability of an event given that another event The probability of an event given that another event has occurred is called a conditional probability or joint probability. has occurred is called a conditional probability or joint probability. A conditional probability is computed as follows : A conditional probability is computed as follows : The conditional probability of A given B is denoted The conditional probability of A given B is denoted by P ( A | B ). by P ( A | B ). The conditional probability of A given B is denoted The conditional probability of A given B is denoted by P ( A | B ). by P ( A | B ). Conditional Probability

19. 19 Slide © 2007 Thomson South-Western. All Rights Reserved Multiplication Law The multiplication law provides a way to compute the The multiplication law provides a way to compute the probability of the intersection of two events. probability of the intersection of two events. The multiplication law provides a way to compute the The multiplication law provides a way to compute the probability of the intersection of two events. probability of the intersection of two events. The law is written as: The law is written as: P ( A  B ) = P ( B ) P ( A | B )

20. 20 Slide © 2007 Thomson South-Western. All Rights Reserved Retrieving information from joint probabilities n The following table illustrates the promotion status of police officers over the past two years. MenWomenTotal Promoted28836324 Not promoted 672204876 Total9602401200 n Based on the information above, calculate all joint probabilities.

21. 21 Slide © 2007 Thomson South-Western. All Rights Reserved Independent Events If the probability of event A is not changed by the If the probability of event A is not changed by the existence of event B, we would say that events A existence of event B, we would say that events A and B are independent. and B are independent. If the probability of event A is not changed by the If the probability of event A is not changed by the existence of event B, we would say that events A existence of event B, we would say that events A and B are independent. and B are independent. Two events A and B are independent if: Two events A and B are independent if: P ( A | B ) = P ( A ) P ( B | A ) = P ( B ) or

22. 22 Slide © 2007 Thomson South-Western. All Rights Reserved The multiplication law also can be used as a test to see The multiplication law also can be used as a test to see if two events are independent. if two events are independent. The multiplication law also can be used as a test to see The multiplication law also can be used as a test to see if two events are independent. if two events are independent. The law is written as: The law is written as: P ( A  B ) = P ( A ) P ( B ) Multiplication Law for Independent Events

23. 23 Slide © 2007 Thomson South-Western. All Rights Reserved Bayes’ Theorem NewInformationNewInformationApplication of Bayes’ TheoremApplication TheoremPosteriorProbabilitiesPosteriorProbabilitiesPriorProbabilitiesPriorProbabilities Often we begin probability analysis with initial or Often we begin probability analysis with initial or prior probabilities. prior probabilities. Then, from a sample, special report, or a product Then, from a sample, special report, or a product test we obtain some additional information. test we obtain some additional information. Given this information, we calculate revised or Given this information, we calculate revised or posterior probabilities. posterior probabilities. Bayes’ theorem provides the means for revising the Bayes’ theorem provides the means for revising the prior probabilities. prior probabilities.

24. 24 Slide © 2007 Thomson South-Western. All Rights Reserved Bayes’ Theorem n Bayes’ theorem is applicable when the events for which we want to compute posterior probabilities are mutually exclusive and their union is the entire sample space.

25. 25 Slide © 2007 Thomson South-Western. All Rights Reserved Bayes Theorem – Two events case n Two events case (A 1 and A 2, mutually exclusive)

26. 26 Slide © 2007 Thomson South-Western. All Rights Reserved Bayes’ Theorem – n events case To find the posterior probability that event A i will To find the posterior probability that event A i will occur given that event B has occurred, we apply occur given that event B has occurred, we apply Bayes’ theorem. Bayes’ theorem.