Chapter 5 A Survey of Probability Concepts Our

Chapter 5 A Survey of Probability Concepts

Our Objectives Define probability. ¢ Describe the classical, empirical, and subjective approaches to probability. ¢ Understand the terms experiment, event, and outcome. ¢ Define the terms conditional probability and joint probability. ¢

Our Objectives (cont’d) Calculate probabilities using the rules of addition and rules of multiplication. ¢ Use a tree diagram to organize and compute probabilities. ¢

Probability A value between zero and one, inclusive, describing the relative possibility (chance or likelihood) an event will occur. ¢ The probability of 1 represents something that is certain to happen; the probability of 0 represents something that cannot happen. ¢

Experiment in Statistics ¢ A process that leads to the occurrence of one and only one of several possible observations.

Outcome in Statistics ¢ A particular result of an experiment. Event in Statistics ¢ A collection of one or more outcomes of an experiment.

Examples Experiment: roll a die. ¢ An outcome: a 6. ¢ Event: observe a number greater than 3. ¢

Approaches to Assigning Probabilities ¢ Objective Probability Classical Probability l Empirical Probability l ¢ Subjective Probability

Classical Probability Based on the assumption that the outcomes of an experiment are equally likely. ¢ Probability of an event= Number of favorable outcomes / Total number of possible outcomes. ¢

Example We roll a die. What is the probability of the event “an even # appears face up”? ¢ Possible outcomes are: 1, 2, 3, 4, 5, 6. (6) ¢ Favorable outcomes are: 2, 4, 6. (3) ¢ Probability of an even number=3/6 =. 5 ¢

Sum of Classical Probabilities If a set of events is mutually exclusive and collectively exhaustive, then the sum of the probabilities is 1. ¢ Mutually exclusive: occurrence of one event means that none of the other events can occur at the same time. ¢ Collectively exhaustive: at least one of the events must occur when an experiment is conducted. ¢

Empirical Probability ¢ Based on relative frequency. ¢ Probability of an event= Number of times event occurred in the past / Total number of observations

Example What is the probability of a future space shuttle mission being successful, given that 2 out of the last 113 missions ended with a disaster? ¢ Probability of a successful mission= Number of successful flights / Total number of flights. ¢ P(A)= 111 / 113=. 98 ¢

Subjective Probability The likelihood of a particular event happening that is assigned by an individual based on whatever information is available. ¢ There is little or no past experience or information on which to base a probability. ¢

Rules for Computing Probabilities ¢ Rules of Addition l Special rule of addition: P(A or B)=P(A) + P(B) (the events must be mutually exclusive)

Example Weight Event Underweight A 100 . 025 Satisfactory B 3, 600 . 900 Overweight C 300 . 075 4, 000 1. 000 Total # of Packages Probability of Occurrence What is the probability that a particular package will be either underweight or overweight?

Example (cont’d) The outcome underweight is event A. ¢ The outcome overweight is event C. ¢ P(A or C)= P(A) + P(C) =. 025 +. 075 =. 10 ¢

Venn Diagram ¢ Graphically portrays the outcome of an experiment. Event A Event B Event C

Complement Rule P(A) + P(~A)= 1 Or P(A)= 1 – P(~A): complement rule Event A ~A

Example Use complement rule to show probability of a satisfactory bag. ¢ Probability of unsatisfactory bag is P(A or C)= P(A) + P(C)=. 100 ¢ Probability of satisfactory bag is P(B)= 1 – [P(A) + P(C)]= 1 –. 100=. 900 ¢ C A. 025 ~(A or C). 900 . 075

General Rule of Addition Used when outcomes of experiment are not mutually exclusive. ¢ P(A or B)= P(A) + P(B) – P(A and B) ¢ A B A and B Joint Probability of A and B

Example What is the probability of a card chosen at random from a deck of cards will be either a king or a heart? ¢ King: P(A)= 4/52 ¢ Heart: P(B) = 13/52 ¢ King of Hearts: P(A and B)= 1/52 ¢ P(A or B)= P(A) + P(B) – P(A and B) ¢ = 4/52 + 13/52 – 1/52 =. 3077

Rules of Multiplication Used to calculate probability of two events happening. ¢ Special rule : ¢ Used when 2 events are independent (occurrence of one has no effect on probability of occurrence of second). l P(A and B)= P(A)P(B) l

Rules of Multiplication (cont’d) ¢ ¢ A survey of AAA revealed 60% of members made airline reservations last year. Two members are selected at random. What is the probability both made reservations last year? 1 st member making reservation: P(R 1)=. 60 2 nd member making reservation: P(R 2)=. 60 P(R 1 and R 2)=P(R 1)P(R 2)=(. 60)=. 36

Rules of Multiplication (cont’d) ¢ If 2 events are dependent we use the general rule of multiplication. ¢ Conditional probability: the probability of a particular event occurring, given that another event has occurred.

General Rule of Multiplication For 2 events, A and B, the joint probability that both will occur is found by multiplying the probability event A will occur, by the conditional probability of event B occurring, given that A has occurred. ¢ P(A and B)= P(A)P(B|A) ¢

Example ¢ ¢ We have a box with 10 rolls of film, of which 3 are defective. What is the probability of getting a defective roll the first time we draw, followed by a defective roll the second time we draw (assuming there are no replacements)? 1 st film being defective: P(D 1)=3/10 (3 out of 10 are defective) 2 nd film being defective: P(D 2|D 1)= 2/9 (now 2 out of 9 are defective) P(D 1 and D 2)= P(D 1)P(D 2|D 1)=(3/10)(2/9)=6/90=. 07

Contingency Tables A table used to classify sample observations according to 2 or more identifiable characteristics. ¢ Is a cross tabulation that simultaneously summarizes 2 or more variables of interest and their relationship. ¢

Example of Contingency Table ATM withdrawals per week 0 Male Female Total 20 40 60 1 40 30 70 2 or more 10 10 20 Total 70 80 150

Example-Contingency Table Application A sample of 200 executives were surveyed about their loyalty to the company. They also indicated their years of service with the company, B 1 being <1 year, B 2 being 1 -5 years, B 3 6 -10 years, and B 4 >10 years. What is the probability of randomly selecting an executive who would remain or has less than 1 year of service?

Example (cont’d) ¢ ¢ ¢ Event A 1: an executive selected at random will remain with the company. P(A 1)= 120/200=. 60 (120 out of 200 would remain) Event B 1: an executive selected at random has <1 year of service. P(B 1)= 35/200=. 175 P(A 1 and B 1)=10/200=. 05 P(A 1 or B 1)= P(A 1)+P(B 1)-P(A 1 and B 1) =. 60 +. 175 -. 05 =. 725

Tree Diagram A graph that is helpful in organizing calculations involving several stages. ¢ The branches of a tree diagram are weighted by probabilities. ¢

Homework ¢ 12 th edition: l ¢ 48, 49, 50 (pg. 171), 52, 54, 55 (pg. 172), 67 (pg. 174). 13 th edition: l 48, 49, 50, 52 (pg. 172), 54, 55 (pg. 173), 66 (pg. 174).