1 Copyright © 1998, Triola, Elementary Statistics Addison Wesley Longman Probability Chapter 3 M A R I O F. T R I O L A Copyright © 1998, Triola, Elementary Statistics, Addison Wesley Longman

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

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

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

5 Copyright © 1998, Triola, Elementary Statistics Addison Wesley Longman  Experiment Toss a fair dice  Events {1} One Example

6 Copyright © 1998, Triola, Elementary Statistics Addison Wesley Longman  Experiment Toss a fair dice  Events {1}, {5} One Example

7 Copyright © 1998, Triola, Elementary Statistics Addison Wesley Longman  Experiment Toss a fair dice  Events {1}, {5}, { 2, 4, 6} One Example

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

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

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

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

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

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

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

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

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

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

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

19 Copyright © 1998, Triola, Elementary Statistics Addison Wesley Longman Notation P - denotes a probability A, B,... - denote a specific event P(A) - denotes the probability of an event occurring

20 Copyright © 1998, Triola, Elementary Statistics Addison Wesley Longman 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

21 Copyright © 1998, Triola, Elementary Statistics Addison Wesley Longman 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 P(A) = number of times A occurred number of times experiment repeated

22 Copyright © 1998, Triola, Elementary Statistics Addison Wesley Longman 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

23 Copyright © 1998, Triola, Elementary Statistics Addison Wesley Longman 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) = number of ways A can occur number of simple events experiment repeated s n

24 Copyright © 1998, Triola, Elementary Statistics Addison Wesley Longman Rule 1 The relative frequency approach is an approximation. Rule 2 The classical approach is the actual probability.

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

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

27 Copyright © 1998, Triola, Elementary Statistics Addison Wesley Longman 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.

28 Copyright © 1998, Triola, Elementary Statistics Addison Wesley Longman Illustration of Law of Large Numbers Proportion of Girls Number of Births Figure

29 Copyright © 1998, Triola, Elementary Statistics Addison Wesley Longman Probability Limits  The probability of an impossible event is 0.

30 Copyright © 1998, Triola, Elementary Statistics Addison Wesley Longman Probability Limits  The probability of an impossible event is 0.  The probability of an event that is certain to occur is 1.

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

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

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

34 Copyright © 1998, Triola, Elementary Statistics Addison Wesley Longman Possible Values for Probabilities Certain Likely Chance Unlikely Impossible Figure

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

36 Copyright © 1998, Triola, Elementary Statistics Addison Wesley Longman Complementary Events The complement of event A, denoted by A, consists of all outcomes in which event A does not occur.

37 Copyright © 1998, Triola, Elementary Statistics Addison Wesley Longman P(A) 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”)

38 Copyright © 1998, Triola, Elementary Statistics Addison Wesley Longman Rounding Off Probabilities  give the exact fraction or decimal

39 Copyright © 1998, Triola, Elementary Statistics Addison Wesley Longman Rounding Off Probabilities  give the exact fraction or decimal or  round off the final result to three significant digits

40 Copyright © 1998, Triola, Elementary Statistics Addison Wesley Longman Subjective Probability A guessed or estimated probability based on knowledge of relevant circumstances.

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

42 Copyright © 1998, Triola, Elementary Statistics Addison Wesley Longman Odds  the odds in favor of event A are the reciprocal of the odds against that event, b:a (or ‘b to a’)  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