probability part 1. l. wang, department of statistics university of south carolina; slide 2 a few...

ProbabilityProbabilityPart 1Part 1

L. Wang, Department of StatisticsL. Wang, Department of Statistics

University of South Carolina; Slide University of South Carolina; Slide 22

A Few TermsA Few Terms

ProbabilityProbability represents a represents a (standardized) measure of chance, and (standardized) measure of chance, and quantifies uncertainty.quantifies uncertainty.

Let Let SS = = sample spacesample space which is the set which is the set of all possible outcomes.of all possible outcomes.

An An eventevent is a set of possible outcomes is a set of possible outcomes that is of interest.that is of interest.

If If AA is an event, then is an event, then P(A)P(A) is the is the probability that event probability that event AA occurs. occurs.



Identify the Sample SpaceIdentify the Sample Space What is the chance that it will rain today?What is the chance that it will rain today?

The number of maintenance calls for an The number of maintenance calls for an old photocopier is twice that for the new old photocopier is twice that for the new photocopier. What is the chance that thephotocopier. What is the chance that the next call will be regarding an old next call will be regarding an old photocopier?photocopier?

If I pull a card out of a pack of 52 cards, If I pull a card out of a pack of 52 cards, what is the chance it’s a spade?what is the chance it’s a spade?



Union and Intersection of Union and Intersection of EventsEvents

The The intersectionintersection of events A and B of events A and B refers to the probability that both refers to the probability that both event A and event B occur.event A and event B occur.

The The unionunion of events A and B refers to of events A and B refers to the probability that event A occurs or the probability that event A occurs or event B occurs or both events, A & B, event B occurs or both events, A & B, occur.occur.

)( BAP

)( BAP



Mutually Exclusive EventsMutually Exclusive Events

Mutually exclusiveMutually exclusive events can not events can not occur at the same time.occur at the same time.

Mutually Exclusive Events

Not Mutually Exclusive Events

S S



A manufacturer of front lights for A manufacturer of front lights for automobiles tests lamps under a high automobiles tests lamps under a high humidity, high temperature humidity, high temperature environment usingenvironment using intensity intensity and and useful lifeuseful life as the responses of as the responses of interest. The following table shows interest. The following table shows the performance of 200 lamps.the performance of 200 lamps.



Probability of the Union of Two Probability of the Union of Two EventsEvents

What is the probability What is the probability that a randomly that a randomly chosen light will have chosen light will have performed Good in performed Good in Useful Life?Useful Life?

Good in Intensity?Good in Intensity? Good in Useful Life Good in Useful Life oror

Good in Intensity?Good in Intensity?

UsefUseful ul LifeLife

IntenInten GoodGood SatSat UnsaUnsatt

TotalTotal

GoodGood 101000

2525 55 131300

SatSat 3535 1010 55 5050

UnsaUnsatt

1010 88 22 2020

TotalTotal 141455

4343 1212 202000



The Union of Two EventsThe Union of Two Events

If events A & B intersect, you have to If events A & B intersect, you have to subtract out the “double count”.subtract out the “double count”.

If events A & B do not intersect (are If events A & B do not intersect (are mutually exclusive), there is no mutually exclusive), there is no “double count”.“double count”.

)()()()( BAPBPAPBAP

)()()( BPAPBAP



What is the probability What is the probability that a randomly chosen that a randomly chosen light will have light will have performed Good in performed Good in Intensity or Intensity or Satisfactorily in Useful Satisfactorily in Useful life?life?

130/20130/20

A.A. 43/20043/200

B.B. 173/200173/200

C.C. 148/200148/200



TotalTotal

GoodGood 101000

2525 55 131300

SatSat 3535 1010 55 5050

UnsaUnsatt

1010 88 22 2020

TotalTotal 141455

4343 1212 202000



What is the probability What is the probability that a randomly chosen that a randomly chosen light will have performed light will have performed Unsatisfactorily in both Unsatisfactorily in both useful life and intensity?useful life and intensity?

A.A. 2/202/20

B.B. 32/20032/200

C.C. 2/2002/200

D.D. 4/2004/200



TotalTotal

GoodGood 101000

2525 55 131300

SatSat 3535 1010 55 5050

UnsaUnsatt

1010 88 22 2020

TotalTotal 141455

4343 1212 202000



Conditional ProbabilityConditional Probability What is the probability What is the probability

that a randomly chosen that a randomly chosen light performed Good in light performed Good in Useful Life?Useful Life?

Good in Intensity.Good in Intensity. Given that a light had Given that a light had

performed Good in performed Good in Useful Life, what is the Useful Life, what is the probability that it probability that it performed Good in performed Good in Intensity?Intensity?



TotalTotal

GoodGood 101000

2525 55 131300

SatSat 3535 1010 55 5050

UnsaUnsatt

1010 88 22 2020

TotalTotal 141455

4343 1212 202000



Conditional ProbabilityConditional Probability

Given that a light had Given that a light had performed Good in performed Good in Intensity, what is the Intensity, what is the probability that it will probability that it will perform Good in perform Good in Useful Life?Useful Life?

A.A. 100/145100/145

B.B. 100/130100/130

C.C. 100/200100/200



TotalTotal

GoodGood 101000

2525 55 131300

SatSat 3535 1010 55 5050

UnsaUnsatt

1010 88 22 2020

TotalTotal 141455

4343 1212 202000



Given that a light had Given that a light had performed Good in performed Good in Intensity, what is the Intensity, what is the probability that it probability that it performed performed Unsatisfactorily in Unsatisfactorily in Useful life?Useful life?

A.A. 5/125/12

B.B. 5/1305/130

C.C. 5/2005/200

D.D. 10/14510/145



TotalTotal

GoodGood 101000

2525 55 131300

SatSat 3535 1010 55 5050

UnsaUnsatt

1010 88 22 2020

TotalTotal 141455

4343 1212 202000



Conditional ProbabilityConditional Probability

The conditional probability of B, The conditional probability of B, given that A has occurred:given that A has occurred:

)(

)()|(

AP

BAPABP



Probability of IntersectionProbability of Intersection

Solving the conditional probability Solving the conditional probability formula for the probability of the formula for the probability of the intersection of A and B:intersection of A and B:

)|()()( ABPAPBAP

)(

)()|(

AP

BAPABP



We purchase 30% of our parts from We purchase 30% of our parts from Vendor A. Vendor A’s defective rate Vendor A. Vendor A’s defective rate is 5%. What is the probability that a is 5%. What is the probability that a randomly chosen part is defective randomly chosen part is defective and from Vendor A?and from Vendor A?

A.A. 0.2000.200B.B. 0.0500.050C.C. 0.0150.015D.D. 0.0300.030



We are manufacturing We are manufacturing widgets. 50% are red, 30% widgets. 50% are red, 30% are white and 20% are blue. are white and 20% are blue. What is the probability that What is the probability that a randomly chosen widget a randomly chosen widget will not be white?will not be white?

A. 0.70 B. 0.50 C. 0.20 D. 0.65A. 0.70 B. 0.50 C. 0.20 D. 0.65



When a computer goes down, there When a computer goes down, there is a 75% chance that it is due to an is a 75% chance that it is due to an overload and a 15% chance that it is overload and a 15% chance that it is due to a software problem. There is due to a software problem. There is an 85% chance that it is due to an an 85% chance that it is due to an overload or a software problem. overload or a software problem. What is the probability that both of What is the probability that both of these problems are at fault?these problems are at fault?

A. 0.11 B. 0.90 C. 0.05 D. 0.20A. 0.11 B. 0.90 C. 0.05 D. 0.20



It has been found that 80% of all It has been found that 80% of all accidents at foundries involve accidents at foundries involve human error and 40% involve human error and 40% involve equipment malfunction. 35% equipment malfunction. 35% involve both problems. If an involve both problems. If an accident involves an equipment accident involves an equipment malfunction, what is the malfunction, what is the probability that there was also probability that there was also human error?human error?

A. 0.3200 B. 0.4375 C. 0.8500 D. A. 0.3200 B. 0.4375 C. 0.8500 D. 0.87500.8750



Suppose there is no Conditional Suppose there is no Conditional Relationship between Useful Life & Relationship between Useful Life & Intensity.Intensity.

What is the probability What is the probability a light performed Good a light performed Good in Intensity?in Intensity?

Given that a light had Given that a light had performed Good in performed Good in Useful Life, what is the Useful Life, what is the probability that it will probability that it will perform Good in perform Good in Intensity?Intensity?



TotalTotal

GoodGood 121288

1616 1616 161600

SatSat 1616 22 22 2020

UnsaUnsatt

1616 22 22 2020

TotalTotal 161600

2020 2020 202000



When , We Say that When , We Say that Events B and A are Events B and A are IndependentIndependent..

)()|( BPABP

The basic idea underlying independence is that information about event A provides no new information about event B. So “given event A has occurred”, doesn’t change our knowledge about the probability of event B occurring.



There are 10 light bulbs in a bag, There are 10 light bulbs in a bag, 2 are burned out.2 are burned out.

If we randomly choose one and If we randomly choose one and test it, what is the probability test it, what is the probability that it is burned out?that it is burned out?

If we set that bulb aside and If we set that bulb aside and randomly choose a second bulb, randomly choose a second bulb, what is the probability that the what is the probability that the second bulb is burned out?second bulb is burned out?



Near IndependenceNear Independence EX: Car company ABC manufactured EX: Car company ABC manufactured

2,000,000 cars in 2008; 1,500,000 of 2,000,000 cars in 2008; 1,500,000 of the cars had anti-lock brakes.the cars had anti-lock brakes.– If we randomly choose 1 car, what If we randomly choose 1 car, what

is the probability that it will have is the probability that it will have anti-lock brakes?anti-lock brakes?

– If we randomly choose another car, If we randomly choose another car, not returning the first, what is the not returning the first, what is the probability that it will have anti-lock probability that it will have anti-lock brakes?brakes?



IndependenceIndependence

Sampling Sampling with replacementwith replacement makes individual selections makes individual selections independent from one another.independent from one another.

Sampling Sampling without replacement without replacement from a very large populationfrom a very large population makes individual selection almost makes individual selection almost independent from one anotherindependent from one another



Probability of IntersectionProbability of Intersection

Probability that both events A and B Probability that both events A and B occur:occur:

If A and B are independent, then the If A and B are independent, then the probability that both occur:probability that both occur:

)|()()( ABPAPBAP

)()()( BPAPBAP



Test for IndependenceTest for Independence

If , then A and B are If , then A and B are independent events.independent events.

If A and B are not independent If A and B are not independent events, they are said to be events, they are said to be dependentdependent events. events.

)()|( BPABP



Four electrical components are Four electrical components are connected in series. The reliability connected in series. The reliability (probability the component operates) (probability the component operates) of each component is 0.90. If the of each component is 0.90. If the components are independent of one components are independent of one another, what is the probability that another, what is the probability that the circuit works when the switch is the circuit works when the switch is thrown?thrown?

A. 0.3600 B. 0.6561 C. 0.7290 D. 0.9000

A B C D



Complementary EventsComplementary Events The The complement of an eventcomplement of an event is is

every outcome not included in the every outcome not included in the event, but still part of the sample event, but still part of the sample space.space.

The complement of event A is The complement of event A is denoted A.denoted A.

Event A is not event A.Event A is not event A. 1)()( APAP

)(1)( APAP

S:

A A

The The complement of an eventcomplement of an event is is every outcome not included in the every outcome not included in the event, but still part of the sample event, but still part of the sample space.space.

The complement of event A is The complement of event A is denoted A.denoted A.

Event A is not event A.Event A is not event A.



Mutually exclusive events Mutually exclusive events are always complementary.are always complementary.

A.A. TrueTrue

B.B. FalseFalse



An automobile manufacturer gives a 5-An automobile manufacturer gives a 5-year/75,000-mile warranty on its drive year/75,000-mile warranty on its drive train. Historically, 7% of the train. Historically, 7% of the manufacturer’s automobiles have manufacturer’s automobiles have required service under this warranty. required service under this warranty. Consider a random sample of 15 cars.Consider a random sample of 15 cars.

If we assume the cars are independent If we assume the cars are independent of one another, what is the probability of one another, what is the probability that no cars in the sample require that no cars in the sample require service under the warrantee?service under the warrantee?

What is the probability that at least one What is the probability that at least one car in the sample requires service?car in the sample requires service?



Consider the following electrical Consider the following electrical circuit:circuit:

The probability on the components is The probability on the components is their reliability (probability that they will their reliability (probability that they will operate when the switch is thrown). operate when the switch is thrown). Components are independent of one Components are independent of one another.another.

What is the probability that the circuit What is the probability that the circuit willwill notnot operate when the switch is operate when the switch is thrown?thrown?

0.95 0.95 0.95



Probability RulesProbability Rules

1)1) 0 0 << P(A) P(A) << 1 1

2)2) Sum of all possible mutually exclusive Sum of all possible mutually exclusive outcomes is 1.outcomes is 1.

3)3) Probability of A or B:Probability of A or B:

4)4) Probability of A or B when A, B are Probability of A or B when A, B are mutually exclusive:mutually exclusive:

)()()()( BAPBPAPBAP

)()()( BPAPBAP



Probability Rules ContinuedProbability Rules Continued

4)4) Probability of B given A:Probability of B given A:

5)5) Probability of A and B:Probability of A and B:

6)6) Probability of A and B when A, B are Probability of A and B when A, B are independent:independent:

)(

)()|(

AP

BAPABP

)|()()( ABPAPBAP

)()()( BPAPBAP



Probability Rules ContinuedProbability Rules Continued

7)7) If A and B are compliments:If A and B are compliments:

1)()( APAP

)(1)( APAP

or



Consider the electrical circuit below. Consider the electrical circuit below. Probabilities on the components are Probabilities on the components are reliabilities and all components are reliabilities and all components are independent. What is the probability independent. What is the probability that the circuit will work when the that the circuit will work when the switch is thrown?switch is thrown?

A

0.90

B

0.90

C

0.95



The number of maintenance calls for The number of maintenance calls for an old photocopier is twice that for an old photocopier is twice that for

the new photocopier.the new photocopier.

A.A. Maintenance Call for Old Machine.Maintenance Call for Old Machine.

B.B. Maintenance Call for New Machine.Maintenance Call for New Machine.

C.C. Two maintenance calls in a row for old machine.Two maintenance calls in a row for old machine.

D.D. Two maintenance calls in a row for new machineTwo maintenance calls in a row for new machine

Outcomes Old Machine New Machine

Probability 0.67 0.33Which of the following series of events would most cause you to question the validity of the above probability model?

probability part 1. l. wang, department of statistics university of south carolina; slide 2 a few...

Documents

event b

exclusive events ss

terms probability

conditional probability

union of events

intersection of events

exclusive eventsnot

set of possible outcomes