Probability A Brief Look. A Few Terms Probability represents a standardized measure of chance, and quantifies uncertainty. Probability represents a standardized

ProbabilityProbabilityA Brief LookA Brief Look

A Few TermsA Few Terms

ProbabilityProbability represents a standardized represents a standardized measure of chance, and quantifies measure of chance, and quantifies uncertainty.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 outcome, then is an event outcome, then P(A)P(A) is is the probability that event outcome the probability that event outcome AA occurs.occurs.

Sample Space, A’s and P(A)’sSample Space, A’s and P(A)’s 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 and

either/or

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

Roommate profile Roommate profile DistributionDistribution

Snores Doesn’t Snore

Parties 150 100 250

Doesn’t Party 200 550 750

350 650 1000

University of South Carolina; Slide University of South Carolina; Slide 66


Parties 0.15 0.10 0.25

Doesn’t Party 0.20 0.55 0.75

0.35 0.65 1.00

Frequency - Counts

Relative Frequency - Probability

What is the probability that What is the probability that a randomly chosen a randomly chosen roommate will snore?roommate will snore?


Parties 150 100 250


350 650 1000

What is the probability that What is the probability that a randomly chosen a randomly chosen roommate will like to party?roommate will like to party?


Parties 150 100 250


350 650 1000

What is the probability that What is the probability that a randomly chosen a randomly chosen roommate will snore roommate will snore oror like like to party?to party?


Parties 150 100 250


350 650 1000

The Union of Two EventsThe Union of Two Events

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

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

)()()()( BAPBPAPBAP

)()()( BPAPBAP

Given that a randomly Given that a randomly chosen roommate snores, chosen roommate snores, what is the probability that what is the probability that he/she likes to party?he/she likes to party?


Parties 150 100 250


350 650 1000



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

Given

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

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 EX: Car company ABC

manufactured 2,000,000 cars in manufactured 2,000,000 cars in 2009; 1,500,000 of the cars had 2009; 1,500,000 of the cars had anti-lock brakes.anti-lock brakes.– If we randomly choose 1 car, If we randomly choose 1 car,

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

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

Are snoring or not and Are snoring or not and partying or not independent partying or not independent of one another?of one another?


Parties 150 100 250


350 650 1000


Arrange the counts so that Arrange the counts so that snoring and partying are snoring and partying are independent of one another.independent of one another.


Parties

Doesn’t Party

1000



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.


All mutually exclusive All mutually exclusive events are complementary.events are complementary.

A.A. TrueTrue

B.B. FalseFalse

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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 A are compliments:If A and A are compliments:

1)()( APAP

)(1)( APAP

or

So Let’s Apply the RulesSo Let’s Apply the Rules

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?


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


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When a computer goes down, there When a computer goes down, there is a 75% chance that it is due to is a 75% chance that it is due to 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?


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



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 B C D

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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 warranty?service under the warranty?

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

A B C


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. Two maintenance calls for an old machine followed Two maintenance calls for an old machine followed by a call for a new machine.by a call for a new machine.

B.B. Two maintenance calls for new machines followed by Two maintenance calls for new machines followed by a call for an old machine.a call for an old machine.

C.C. Three maintenance calls in a row for an old machine.Three maintenance calls in a row for an old machine.D.D. Three maintenance calls in a row for a new machineThree maintenance calls in a row for a 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?


??????????

What is the probability that at least 2 What is the probability that at least 2 people in this class (n=39) have the people in this class (n=39) have the same birthday – Month and day?same birthday – Month and day?– Year has 365 days – forget leap year.Year has 365 days – forget leap year.– Equally likelihood for each dayEqually likelihood for each day

Multiplication of ChoicesMultiplication of Choices

If an operation can be performed in If an operation can be performed in nn11 ways, and if for each of these a ways, and if for each of these a second operation can be performed in second operation can be performed in nn22 ways, and for each of the first two a ways, and for each of the first two a third operation can be performed in third operation can be performed in nn33 ways, and so forth, then the sequence ways, and so forth, then the sequence of of kk operations can be performed in operations can be performed in nn11 ·· nn22 ··……· · nnkk ways. ways.


Multiplication of ChoicesMultiplication of Choices There are 5 processes needed to There are 5 processes needed to

manufacture the side panel for a car: clean, manufacture the side panel for a car: clean, press, cut, paint, polish. Our plant has 6 press, cut, paint, polish. Our plant has 6 cleaning stations, 3 pressing stations, 8 cleaning stations, 3 pressing stations, 8 cutting stations, 5 painting stations, and 8 cutting stations, 5 painting stations, and 8 polishing stations.polishing stations.– How many different “pathways” through the How many different “pathways” through the

manufacturing exist?manufacturing exist?– What is the number of “pathways” that include a What is the number of “pathways” that include a

particular pressing station?particular pressing station?– What is the probability that a panel follows any What is the probability that a panel follows any

particular path?particular path?– What is the probability that a panel goes through What is the probability that a panel goes through

pressing station 1?pressing station 1?

Classical Definition of Classical Definition of ProbabilityProbability


Nn

AP )(

If an experiment can result in any one of N different, but equally likely, outcomes, and if exactly n of these outcomes corresponds to event A, then the probability of event A is

CountingCounting

Suppose there are 3 vendors and we Suppose there are 3 vendors and we want to choose 2. How many want to choose 2. How many possible combinations of 2 can be possible combinations of 2 can be chosen from the 3 vendors?chosen from the 3 vendors?


Counting - CombinationsCounting - Combinations


The number of combinations of The number of combinations of nn distinct objects taken distinct objects taken rr at a time is at a time is

)!(!

!

rnr

n

r

nCrn

“n choose r”

Factorial ReminderFactorial Reminder

n! = n n! = n · · (n-1) (n-1) ·· (n-2) (n-2) ··…..…..– EX: 4! = (4)(3)(2)(1) = 24EX: 4! = (4)(3)(2)(1) = 24

1! = 1 0! = 11! = 1 0! = 1


10)!25(!2

!5

2

525

C

CountingCounting 9 out of 100 computer chips are defective. 9 out of 100 computer chips are defective.

We choose a random sample of n=3.We choose a random sample of n=3.– How many different samples of 3 are possible?How many different samples of 3 are possible?

– How many of the samples of 3 contain exactly 1 How many of the samples of 3 contain exactly 1 defective chip? defective chip?

– What is the probability of choosing exactly 1 What is the probability of choosing exactly 1 defective chip in a random sample of 3?defective chip in a random sample of 3?

– What is the probability of choosing at least 1 What is the probability of choosing at least 1 defective chip in a random sample of 3? defective chip in a random sample of 3?


This class consists of 5 women and This class consists of 5 women and 34 men. If we randomly choose 4 34 men. If we randomly choose 4 people, what is the probability that people, what is the probability that there will be no women chosen?there will be no women chosen?

What is the probability that there will What is the probability that there will be at least 1 woman chosen?be at least 1 woman chosen?


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Probability A Brief Look. A Few Terms Probability represents a standardized measure of chance, and quantifies uncertainty. Probability represents a standardized