Probability Chapter 3 M A R I O

Chapter 3 Probability 3 -1 Overview* 3 -2 Fundamentals 3 -3 Addition Rule 3 -4 & 5 Multiplication Rule 3 -6 Probabilities Through Simulations* 3 -7 Counting * Reading Material Copyright © 1998, Triola, Elementary Statistics Addison Wesley Longman 2

Chapter 3 Overview Objectives v develop sound understanding of probability values used in subsequent chapters v develop basic skills necessary to solve simple probability problems Copyright © 1998, Triola, Elementary Statistics Addison Wesley Longman 3

3 -2 Fundamentals Definitions v Experiment - any process to obtain observations v Event - any collection of results or outcomes of an experiment v Simple event - any outcome or event that cannot be broken down any further v Sample space - all possible simple events Copyright © 1998, Triola, Elementary Statistics Addison Wesley Longman 4

One Example v Experiment Toss a fair dice v Events {1} Copyright © 1998,

One Example v Experiment Toss a fair dice v Events {1}, {5} Copyright ©

One Example v Experiment Toss a fair dice v Events {1}, {5}, { 2,

One Example v Experiment Toss a fair dice v Events {1}, {5}, { 2, 4, 6}, {2, 3}, {1, 2, 3, 5} v Simple events {1}, {2}, {3}, {4} Copyright © 1998, Triola, Elementary Statistics Addison Wesley Longman 14

One Example v Experiment Toss a fair dice v Events {1}, {5}, { 2, 4, 6}, {2, 3}, {1, 2, 3, 5} v Simple events {1}, {2}, {3}, {4}, {5} Copyright © 1998, Triola, Elementary Statistics Addison Wesley Longman 15

One Example v Experiment Toss a fair dice v Events {1}, {5}, { 2, 4, 6}, {2, 3}, {1, 2, 3, 5} v Simple events {1}, {2}, {3}, {4}, {5}, {6} Copyright © 1998, Triola, Elementary Statistics Addison Wesley Longman 16

One Example v Experiment Toss a fair dice v Events {1}, {5}, { 2, 4, 6}, {2, 3}, {1, 2, 3, 5} v Simple events {1}, {2}, {3}, {4}, {5}, {6} v Sample space Copyright © 1998, Triola, Elementary Statistics Addison Wesley Longman 17

One Example v Experiment Toss a fair dice v Events {1}, {5}, { 2, 4, 6}, {2, 3}, {1, 2, 3, 5} v Simple events {1}, {2}, {3}, {4}, {5}, {6} v Sample space {1, 2, 3, 4, 5, 6} Copyright © 1998, Triola, Elementary Statistics Addison Wesley Longman 18

Notation P - denotes a probability A, B, . . . - denote a

Two Basic Rules for Computing Probability Rule 1: Relative frequency approximation Conduct an experiment a large number of times and count the number of times event A actually occurs, then the estimate of P(A) is Copyright © 1998, Triola, Elementary Statistics Addison Wesley Longman 20

Two Basic Rules for Computing Probability Rule 1: Relative frequency approximation Conduct an experiment a large number of times and count the number of times event A actually occurs, then the estimate of P(A) is number of times A occurred P(A) = number of times experiment repeated Copyright © 1998, Triola, Elementary Statistics Addison Wesley Longman 21

Two Basic Rules for Computing Probability Rule 2: Classical approach If experiment has n different simple events, each with an equal chance of occurring, and s is the number of ways event A can occur, then Copyright © 1998, Triola, Elementary Statistics Addison Wesley Longman 22

Two Basic Rules for Computing Probability Rule 2: Classical approach If experiment has n different simple events, each with an equal chance of occurring, and s is the number of ways event A can occur, then P(A) = s n number of ways A can occur number of simple events experiment repeated Copyright © 1998, Triola, Elementary Statistics Addison Wesley Longman 23

Rule 1 The relative frequency approach is an approximation. Rule 2 The classical approach

Example: Toss a fair coin Rule 1: Toss the coin 100 times, the face comes up 47 times P(face up) = 47/100 =. 47 Rule 2: Two simple events, face up or face down, equal chance P(face up) = 1/2 =. 5 Copyright © 1998, Triola, Elementary Statistics Addison Wesley Longman 25

Example: Toss a fair dice Rule 1: Toss the dice 100 times, face 1 comes up 18 times P(face up) = 18/100 =. 18 Rule 2: Six simple events, {1}, {2} {3}, {4}, {5} & {6}, equal chance P(face up) = 1/6 =. 16666 Copyright © 1998, Triola, Elementary Statistics Addison Wesley Longman 26

Law of Large Numbers As an experiment is repeated again and again, the relative frequency probability (from Rule 1) of an event tends to approach the actual probability. Copyright © 1998, Triola, Elementary Statistics Addison Wesley Longman 27

Illustration of Law of Large Numbers Proportion of Girls 0. 6 0. 5 • • • • 0. 4 0. 3 • • • • 0. 2 0. 1 0 20 40 60 80 100 120 Number of Births Figure Copyright © 1998, Triola, Elementary Statistics Addison Wesley Longman 28

Probability Limits v The probability of an impossible event is 0. Copyright © 1998,

Probability Limits v The probability of an impossible event is 0. v The probability of an event that is certain to occur is 1. 0 £ P(A) £ 1 Copyright © 1998, Triola, Elementary Statistics Addison Wesley Longman 31

Probability Limits v The probability of an impossible event is 0. v The probability of an event that is certain to occur is 1. 0 £ P(A) £ 1 Impossible to occur Copyright © 1998, Triola, Elementary Statistics Addison Wesley Longman 32

Probability Limits v The probability of an impossible event is 0. v The probability of an event that is certain to occur is 1. 0 £ P(A) £ 1 Impossible to occur Certain to occur Copyright © 1998, Triola, Elementary Statistics Addison Wesley Longman 33

Possible Values for Probabilities Certain 1 Likely 0. 5 50 -50 Chance Unlikely Figure

Complementary Events Copyright © 1998, Triola, Elementary Statistics Addison Wesley Longman 35

Complementary Events The complement of event A, denoted by A, consists of all outcomes in which event A does not occur. P(A) (read “not A”) Copyright © 1998, Triola, Elementary Statistics Addison Wesley Longman 37

$Rounding Off Probabilities v give the exact fraction or decimal or v round off$

Subjective Probability A guessed or estimated probability based on knowledge of relevant circumstances. Copyright

Odds Copyright © 1998, Triola, Elementary Statistics Addison Wesley Longman 41

Odds v the odds against event A occurring are the ratio P(A) / P(A), usually expressed in the form of a: b (or ‘a to b’), where a and b are integers with no common factors v the odds in favor of event A are the reciprocal of the odds against that event, b: a (or ‘b to a’) Copyright © 1998, Triola, Elementary Statistics Addison Wesley Longman 42