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INSE 6320 -- Week 2

Risk Analysis for Information and Systems Engineering

Risk and UncertaintyElementary Probability Theory

Dr. A. Ben Hamza Concordia University

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Statisticians Probabilities

Consequences of Adverse Events

Quantifiable

Social scientists Invented to cope with uncertainties

Dependent on perception

Risk perception: blending of science and judgment with importantpsychological, social, cultural, and political factors

Risk estimation depends on risk definition

RiskOpposing Views (Life is risky. The future is uncertain).

Needs to be a consistent and universallyaccepted definition of risk per domain

Our risk domain is information security

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Uncertainty in computing risk is unavoidable Reactions to risk based on emotion, rather than scientific evidence.

When people become outraged, they may overreact.

If people are not outraged, they may under-react.

An industrial process producing an unpronounceable chemical is a much lessacceptable risk than something more everyday, like driving or eating junk food.

Risk comparisons may be more clear than using absolute numbers Emotions must be considered with scientific evidence.

RiskHuman Factors

People become uneasy when scientists arenot certain about the risk posed by a hazard(effect, severity, or prevalence).

Rather than diminish legitimate concernsor heighten illegitimate ones,psychological factors must be addressedto encourage constructive action.

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Risk can be viewed as uncertainty and similarly risk analysis can beviewed as decision making in terms of uncertainty.

Risk be analyzed intuitively or analytically In a lot of day to day activities risk is considered intuitively

Such skills are honed via years of experience in dealing with some situations

Humans have limitations in handling multiple pieces of information Analytic techniques are required for complex problems where a lot of factors

are required.

RiskSummary

A man with a watch knows what time it is. A man with twowatches is never sure

(Unknown)

Uncertainty in Measurements

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Types of Probability

Classical (theoretical) Probability

Each outcome in a sample space is equally likely.

Number of outcomes in event E( )

Number of outcomes in sample spaceP E

Empirical (statistical) Probability

Based on observations obtained from probability experiments. Relative frequency of an event.

Frequency of event E( )

Total frequency

fP E

n

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Example: Finding Classical Probabilities

1. Event A: rolling a 3

2. Event B: rolling a 73. Event C: rolling a number less than 5

Solution:

Sample space: {1, 2, 3, 4, 5, 6}

You ro ll a six-sided die. Find the probability of each event.

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Solution: Finding Classical Probabilities

1. Event A: rolling a 3 Event A = {3}

1(rolling a 3) 0.167

6P

2. Event B: rolling a 7 Event B= { } (7 is not in the samplespace)

0(rolling a 7) 0

6P

3. Event C: rolling a number less than 5

Event C = {1, 2, 3, 4}

4(rolling a number less than 5) 0.667

6P

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Example: Finding Empirical Probabilities

A company is conducting an online survey of randomly selected individuals todetermine if traffic congestion is a problem in their community. So far, 320people have responded to the survey. What is the probability that the next

person that responds to the survey says that traffic congestion is a seriousproblem in their community?

Response Number of times,f

Serious problem 123

Moderate problem 115

Not a problem 82

f= 320

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Solution: Finding Empirical Probabilities

Response Number of times,f

Serious problem 123

Moderate problem 115

Not a problem 82

f = 320

event frequency

123(Serious problem) 0.384

320

fP

n

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Example: Probability of the Complement of an Event

You survey a sample of 1000 employees at a company and recordthe age of each. Find the probability of randomly choosing anemployee who is not between 25 and 34 years old.

Employee ages Frequency,f

15 to 24 54

25 to 34 366

35 to 44 233

45 to 54 180

55 to 64 125

65 and over 42f= 1000

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Solution: Probability of the Complement of an Event

Use empirical probability to findthe probability P(age 25 to 34) Employee ages Frequency,f

15 to 24 54

25 to 34 366

35 to 44 233

45 to 54 180

55 to 64 125

65 and over 42

f= 1000

366

(age 25 to 34) 0.3661000

f

P n

Use the complement rule

366(age is not 25 to 34) 1

1000

6340.634

1000

P

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Mutually Exclusive Events

Mutually exclusive

Two eventsA andB cannot occur at the same time

A

BA B

A andB are mutually

exclusive

A andB are not mutually

exclusive

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Example: Mutually Exclusive Events

Decide if the events are mutually exclusive.

EventA: Roll a 3 on a die.

EventB: Roll a 4 on a die.

Solution:

Mutually exclusive (The first event has oneoutcome, a 3. The second event also has oneoutcome, a 4. These outcomes cannot occur at thesame time.)

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Example: Mutually Exclusive Events

Decide if the events are mutually exclusive.

EventA: Randomly select a male student.

EventB: Randomly select a nursing major.

Solution:

Not mutually exclusive (The student can be a malenursing major.)

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Computing Joint and Marginal Probabilities

The probability of a joint event,A andB:

Computing a marginal (or simple) probability:

whereB1, B2, , Bkare kmutually exclusive and collectivelyexhaustive events

number of outcomes satisfying and( and ) ( )

total number of elementary outcomes

A BP A B P A B

1 2 kP(A) P(A B ) P(A B ) P(A B )

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Joint Probability Example

P(Red andAce)

Black

ColorType Red Total

Ace 2 2 4

Non-Ace 24 24 48

Total 26 26 52

522

cardsofnumbertotalaceandredarethatcardsofnumber

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Marginal Probability Example

P(Ace)

Black

ColorType Red Total

Ace 2 2 4

Non-Ace 24 24 48

Total 26 26 52

52

4

52

2

52

2)BlackandAce(P)dReandAce(P

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P(A1 and B2) P(A1)

TotalEvent

Marginal & Joint Probabilities In A Contingency Table

P(A2 and B1)

P(A1 and B1)

Event

Total 1

Joint Probabilities Marginal (Simple) Probabil ities

A1

A2

B1 B2

P(B1) P(B2)

P(A2 and B2) P(A2)

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General Addition Rule

General Addit ion Rule:

If A and B are mutually exclusive, then

, so the rule can be simpli fied:

For mutually exclusive events A and B

( ) ( ) ( ) ( )P A B P A P B P A B

( ) 0P A B

( ) ( ) ( )P A B P A P B

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General Addition Rule Example

P(Red orAce) = P(Red) +P(Ace) - P(Red andAce)

= 26/52 + 4/52 - 2/52 = 28/52

Dont count

the two red

aces twice!Black

ColorType Red Total

Ace 2 2 4

Non-Ace 24 24 48

Total 26 26 52

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Independence of Events and Marginal Probability

Two events are independent if and only if: EventsA andB are independent when the probability of one event

is not affected by the fact that the other event has occurred

( | ) ( )P A B P A

The occurrence of one of the events does not affect the probability ofthe occurrence of the other event

P(B |A) = P(B) or P(A |B) = P(A)

Events that are not independent are dependent

Marginal probability for eventA (also called total probability ofA):

whereB1, B

2, , B

karekmutually exclusive and collectively exhaustive events

1 1 2 2( ) ( | ) ( ) ( | ) ( ) ( | ) ( )k kP A P A B P B P A B P B P A B P B

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Multiplication Rules

Multiplication rule for two events A and B:

P(AB) P(A|B) P(B)

P(A)B)|P(A Note: If A and B are independent, thenand the multiplication rule simplifies to

P(AB) P(A)P(B)

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Example: Using the Multiplication Rule

Two cards are selected, without replacing the first card, from astandard deck. Find the probability of selecting a king (K) andthen selecting a queen (Q).

Solution:Because the first card is not replaced, the events are dependent.

( ) ( ) ( | )

4 4

52 51

160.006

2652

P KQ P K P Q K

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Example: Using the Multiplication Rule

A coin is tossed and a die is rolled. Find the probability of getting a headand then rolling a 6.

Solution:The outcome of the coin does not affect the probability of rolling a 6 onthe die. These two events are independent.

( 6) ( ) (6)

1 1

2 6

1 0.08312

P H and P H P

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Example: Using the Addition Rule

You select a card from a standard deck. Find the probability that thecard is a 4 or an ace.

Solution:The events are mutually exclusive (if the card is a 4, it cannot be an ace)

(4 ) (4) ( )

4 4

52 52

80.154

52

P or ace P P ace

4

4

4

4 A

A

A

A

44 other cards

Deck of 52 Cards

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The frequency distribution shows thevolume of sales (in dollars) and the number

of months a sales representative reachedeach sales level during the past threeyears. If this sales pattern continues, whatis the probability that the salesrepresentative will sell between $75,000and $124,999 next month?

Sales volume ($) Months

024,999 3

25,00049,999 5

50,00074,999 6

75,00099,999 7

100,000124,999 9

125,000149,999 2

150,000174,999 3

175,000199,999 1

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Solution: Using the Addition Rule

A = monthly sales between $75,000 and$99,999

B = monthly sales between $100,000and $124,999

A andB are mutually exclusive

Sales volume ($) Months

024,999 3

25,00049,999 5

50,00074,999 675,00099,999 7

100,000124,999 9

125,000149,999 2

150,000174,999 3

175,000199,999 1

( ) ( ) ( )

7 9

36 36

160.444

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P A or B P A P B

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A blood bank catalogs the types of blood given by donors during thelast five days. A donor is selected at random. Find the probability thedonor has type O or type A blood.

Type O Type A Type B Type AB Total

Rh-Positive 156 139 37 12 344

Rh-Negative 28 25 8 4 65

Total 184 164 45 16 409

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