11

ProbabilityProbability

22

Chapter GoalsChapter GoalsAfter completing this chapter, you After completing this chapter, you

should be able to:should be able to: • Explain basic probability concepts Explain basic probability concepts

and definitionsand definitions• Examine the use of probability theory Examine the use of probability theory

in decision makingin decision making• Apply common rules of probabilityApply common rules of probability• Use Bayes’ Theorem for conditional Use Bayes’ Theorem for conditional

probabilitiesprobabilities

33

Important TermsImportant Terms

• ProbabilityProbability – the chance that an – the chance that an uncertain event will occur (always uncertain event will occur (always between 0 and 1)between 0 and 1)

• EventEvent – Each possible type of – Each possible type of occurrence or outcomeoccurrence or outcome

• Sample SpaceSample Space – the collection of all – the collection of all possible eventspossible events

44

Sample SpaceSample SpaceThe Sample Space is the collection of all possible events

e.g. All 6 faces of a die:

e.g. All 52 cards of a bridge deck:

55

EventsEvents• Simple eventSimple event

– An outcome from a sample space with one An outcome from a sample space with one characteristiccharacteristic

– e.g., A red card from a deck of cardse.g., A red card from a deck of cards• Complement of an event A (denoted A’)Complement of an event A (denoted A’)

– All outcomes that are not part of event AAll outcomes that are not part of event A– e.g., All cards that are not diamondse.g., All cards that are not diamonds

• Joint eventJoint event– Involves two or more characteristics Involves two or more characteristics

simultaneouslysimultaneously– e.g., An ace that is also red from a deck of e.g., An ace that is also red from a deck of

cardscards

66

Mutually Exclusive Mutually Exclusive EventsEvents

• Mutually exclusiveMutually exclusive events events– Events that cannot occur togetherEvents that cannot occur together

example:example:

A = queen of diamonds; B = queen of clubsA = queen of diamonds; B = queen of clubs

– Events A and B are mutually Events A and B are mutually exclusiveexclusive

77

Questions:Questions:• Which of the following pairs of mutually Which of the following pairs of mutually

exclusive events in the drawing of one exclusive events in the drawing of one card from a standard deck of 52?card from a standard deck of 52?

a)a) A heart and a queenA heart and a queen

b)b) An even number and a spadeAn even number and a spade

c)c) An ace and an even numberAn ace and an even number

88

Collectively Exhaustive Collectively Exhaustive EventsEvents

• Collectively exhaustiveCollectively exhaustive events events– The set of events covers the entire sample The set of events covers the entire sample

spacespace

example:example: A = aces; B = black cards; A = aces; B = black cards;

C = diamonds; D = heartsC = diamonds; D = hearts

– Events A, B, C and D are collectively Events A, B, C and D are collectively exhaustive (but not mutually exclusive – an exhaustive (but not mutually exclusive – an ace may also be a heart)ace may also be a heart)

– Events B, C and D are collectively exhaustive Events B, C and D are collectively exhaustive and also mutually exclusiveand also mutually exclusive

99

ProbabilityProbability• Probability is the numerical Probability is the numerical

measure of the likelihood that measure of the likelihood that an event will occuran event will occur

• The probability of any event The probability of any event must be between 0 and 1, must be between 0 and 1,

Certain

Impossible

.5

1

0

0 ≤ P(A) ≤ 1 For any event A

1010

Types of ProbabilityTypes of Probability

1.1. Classical approachClassical approach2.2. Relative Frequency approachRelative Frequency approach3.3. Subjective approachSubjective approach

1111

Questions:Questions:• Determine the probabilities of the Determine the probabilities of the

following events in drawing a card from a following events in drawing a card from a deck of 52 cards:deck of 52 cards:

a)a) A sevenA sevenb)b) A black cardA black cardc)c) An ace or a kingAn ace or a kingd)d) A black two or a black threeA black two or a black threee)e) A red face card ( king, queen or jack)A red face card ( king, queen or jack)

Which type of probability estimates are these?Which type of probability estimates are these?

1212

Question:Question:• General Buck Turgidson is preparing to make his annual budget General Buck Turgidson is preparing to make his annual budget

presentation to the U.S. Senate and is speculating about his presentation to the U.S. Senate and is speculating about his chances of getting all or part of his requested budget approved. chances of getting all or part of his requested budget approved. From his 20 years of experience in making these requests, he From his 20 years of experience in making these requests, he has deduced that his chances of getting between 50 to 74 has deduced that his chances of getting between 50 to 74 percent of his budget approved are twice as good as those of percent of his budget approved are twice as good as those of getting between 75 and 99 percent approved, and two and getting between 75 and 99 percent approved, and two and one-half times as good as those of getting between 25 and 49 one-half times as good as those of getting between 25 and 49 percent approved. Further, the general believes that there is no percent approved. Further, the general believes that there is no chance of less than 25 percent of his budget being approved. chance of less than 25 percent of his budget being approved. Finally, the entire budget has been approved only once during Finally, the entire budget has been approved only once during the general’s tenure, and the general does not expect this the general’s tenure, and the general does not expect this pattern to change. What are the probabilities of 0-24 percent, pattern to change. What are the probabilities of 0-24 percent, 50-74 percent, 75-99 percent, and 100 percent approval, 50-74 percent, 75-99 percent, and 100 percent approval, according to the general?according to the general?

1313

Probability RulesProbability Rules• Marginal ProbabilityMarginal Probability

• Addition rule for Mutually Exclusive Addition rule for Mutually Exclusive EventEvent P (A or B) = P(A) + P(B)P (A or B) = P(A) + P(B)

• Addition Rule for Events that are not Addition Rule for Events that are not mutually exclusivemutually exclusive

P(A or B) = P (A) + P(B) – P(AB)P(A or B) = P (A) + P(B) – P(AB)

1414

General Addition Rule General Addition Rule ExampleExample

P(Red or Ace) = P(Red) +P(Ace) - P(Red and Ace)

= 26/52 + 4/52 - 2/52 = 28/52Don’t count the two red aces twice!

BlackColor

Type Red Total

Ace 2 2 4Non-Ace 24 24 48Total 26 26 52

1515

Question:Question: The Herr - Mc Fee Company, which The Herr - Mc Fee Company, which

produces nuclear fuel rods, must X-ray and produces nuclear fuel rods, must X-ray and inspect each rod before shipping. Karen inspect each rod before shipping. Karen Wood, an inspector, has noted that for Wood, an inspector, has noted that for every 1000 fuel rods she inspects, 10 have every 1000 fuel rods she inspects, 10 have interior flaws, 8 have casing flaws, and 5 interior flaws, 8 have casing flaws, and 5 have both flaws. In her quarterly report, have both flaws. In her quarterly report, Karen must include the probability of flaws Karen must include the probability of flaws in fuel rods. What is this probability?in fuel rods. What is this probability?

1616

Question:Question:• The HAL corporation wishes to improve the resistance of its The HAL corporation wishes to improve the resistance of its

personal computer to disk-drive and keyboard failures. At personal computer to disk-drive and keyboard failures. At present, the design of the computer is such that disk drive present, the design of the computer is such that disk drive failures occur only one-third as often a keyboard failures. failures occur only one-third as often a keyboard failures. The probability of simultaneous disk-drive and keyboard The probability of simultaneous disk-drive and keyboard failures is 0.05.failures is 0.05.

a)a) If the computer is 80% resistant to disk-drive and / or If the computer is 80% resistant to disk-drive and / or keyboard failure, how low must the disk–drive failure keyboard failure, how low must the disk–drive failure probability be?probability be?

b)b) If the keyboard is improved so that it fails only twice as If the keyboard is improved so that it fails only twice as often as the disk-drive (and the simultaneous failure often as the disk-drive (and the simultaneous failure probability is still 0.05), will the disk-drive failure probability is still 0.05), will the disk-drive failure probability from part (a) yield a resistance to disk-drive probability from part (a) yield a resistance to disk-drive and/ or keyboard failure higher or lower than 90 percent? and/ or keyboard failure higher or lower than 90 percent?

1717

Probabilities under Probabilities under conditions of Statistical conditions of Statistical

IndependenceIndependence• Marginal ProbabiltyMarginal Probabilty

Simple probability of occurrence of an event.Simple probability of occurrence of an event.

• Joint ProbabilityJoint Probability P(AB) = P(A) X P(B)P(AB) = P(A) X P(B) Probability treeProbability tree

• Conditional probabilityConditional probability P(BP(B|A) = P(B)|A) = P(B)

1818

Question:Question:• The four floodgates of a small hydroelectric The four floodgates of a small hydroelectric

dam fail and are repaired independently of dam fail and are repaired independently of each other. From experience, it’s known that each other. From experience, it’s known that each floodgate is out of order 4 percent of the each floodgate is out of order 4 percent of the time.time.

a)a) If flood gate 1 is out of order, what is the probability If flood gate 1 is out of order, what is the probability that floodgates 2 and 3 are out of order?that floodgates 2 and 3 are out of order?

b)b) During a tour of the dam, you are told that the During a tour of the dam, you are told that the chances of all four floodgates being out of order are chances of all four floodgates being out of order are less than 1 in 5,000,000. Is this statement true?less than 1 in 5,000,000. Is this statement true?

1919

Question:Question:• Rob Rales is preparing a report that his employer, the Titre Rob Rales is preparing a report that his employer, the Titre

Corporation, will eventually deliver to the Federal aviation Corporation, will eventually deliver to the Federal aviation Administration. First, the report must be approved by Rob’s Administration. First, the report must be approved by Rob’s group leader, departmental head, and division chief (in that group leader, departmental head, and division chief (in that order). Rob knows from experience that the three managers order). Rob knows from experience that the three managers act independently. Further, he knows that his group leader act independently. Further, he knows that his group leader approves 85 % of his reports, his departmental head approves 85 % of his reports, his departmental head approves 80% of the reports written by Rob that reach him, approves 80% of the reports written by Rob that reach him, and the division chief approves 82 % of Rob’s work.and the division chief approves 82 % of Rob’s work.

a)a) What is the probability that the first version of Rob’s report is What is the probability that the first version of Rob’s report is submitted to the FAA?submitted to the FAA?

b)b)What is the probability that the first version of Rob’s report is What is the probability that the first version of Rob’s report is approved by his group leader and his departmental head, approved by his group leader and his departmental head, but is not approved by his division chief?but is not approved by his division chief?

2020

Probabilities under Probabilities under conditions of Statistical conditions of Statistical

DependenceDependence• Conditional ProbabilityConditional Probability

P (B P (B l A) = P (BA) / P (A)l A) = P (BA) / P (A)

• Joint ProbabilityJoint Probability P (BA) = P (BA) = P (B P (B l A) x P(A)l A) x P(A)

= = P (A P (A l B) x P(B)l B) x P(B)

• Marginal ProbabiltyMarginal Probabilty– Sum of probabilities of all the joint events in Sum of probabilities of all the joint events in

which the simple event occurs.which the simple event occurs.

2121

Question:Question:• During a study of auto accidents, the During a study of auto accidents, the

Highway Safety Council found that 60 % of Highway Safety Council found that 60 % of all accide4nts occur at night, 52% are all accide4nts occur at night, 52% are alcohol-related, and 37 % occur at night and alcohol-related, and 37 % occur at night and are alcohol related.are alcohol related.

a)a) What is the probability that an accident was What is the probability that an accident was alcohol related, given that it occurred at night?alcohol related, given that it occurred at night?

b)b) What is the probability that an accident occurred What is the probability that an accident occurred at night, given that it was alcohol-related?at night, given that it was alcohol-related?

2222

Question:Question:• If a hurricane forms in the eastern half of the If a hurricane forms in the eastern half of the

Gulf of Mexico, there is a 76% chance that it will Gulf of Mexico, there is a 76% chance that it will strike the western coast of Florida. From data strike the western coast of Florida. From data gathered over the past 50 years, it has been gathered over the past 50 years, it has been determined that the probability of a hurricane’s determined that the probability of a hurricane’s occurring in this area in any given year is 0.85.occurring in this area in any given year is 0.85.

a)a) What is the probability that a hurricane will occur in What is the probability that a hurricane will occur in the eastern Gulf of Mexico and strike Florida this year?the eastern Gulf of Mexico and strike Florida this year?

b)b) If a hurricane in the eastern Gulf of Mexico is seeded If a hurricane in the eastern Gulf of Mexico is seeded (induced to rain by addition of chemicals from (induced to rain by addition of chemicals from aircraft), its probability of striking Florida’s west coast aircraft), its probability of striking Florida’s west coast is reduced by one-fourth. If it is decided to seed any is reduced by one-fourth. If it is decided to seed any hurricane in the eastern gulf, what is the value of hurricane in the eastern gulf, what is the value of probability in part (a)?probability in part (a)?

2323

Bayes’ TheoremBayes’ Theorem• Posterior probabilitiesPosterior probabilities• Basic formula for conditional probability Basic formula for conditional probability

under statistical dependence is called under statistical dependence is called Bayes’ theorem.Bayes’ theorem.

P (B P (B l A) = P (BA) / P (A)l A) = P (BA) / P (A)

2424

Chapter SummaryChapter Summary• Discussed basic probability conceptsDiscussed basic probability concepts

– Sample spaces and events, simple probability, Sample spaces and events, simple probability, and joint probabilityand joint probability

• Examined basic probability rulesExamined basic probability rules– General addition rule, addition rule for mutually General addition rule, addition rule for mutually

exclusive events, rule for collectively exhaustive exclusive events, rule for collectively exhaustive eventsevents

• Defined conditional probabilityDefined conditional probability– Statistical independence, marginal probability, Statistical independence, marginal probability,

decision treesdecision trees• Discussed Bayes’ theoremDiscussed Bayes’ theorem