1

Basic probability theory

Professor Jørn Vatn

2

Event

Probability relates to events Let as an example A be the event that there is an operator

error in a control room next year, and B be the event that there is a specific component failure next year i.e.: A = {operator error next year} B = {component failure next year}

An event may occur, or not. We do not know that in advance prior to the experiment or a situation in the “real life”.

3

Probability

When events are defined, the probability that the event will occur is of interest

Probability is denoted by Pr(·), i.e. Pr(A) = Probability that A (will) occur The numeric value of Pr(A) may be found by:

Studying the sample space / symmetric considerations Analysing collected data Look up the value in data hand books “Expert judgement” Laws of probability calculus/Monte Carlo simulation

4

Sample space

The sample space defines all possible events As an example let A = {It is Sunday}, B = {It is Monday}, .. ,

G = {It is Saturday}. The sample space is then given by S = {A,B,C,D,E,F,G} So-called Venn diagrams are useful when we want to

analyze subset of the sample space S.

5

Venn diagram

A rectangle represents the sample space, and closed curves such as a circle are used to represent subsets of the sample space

SA

6

Union

The union of two events A and B: A B denotes the occurrence of A or B or (A and B)

Example A = {prime numbers 6) B = {odd numbers 6} A B = {1,2,3,5}

S A B

7

Intersection

The intersection of two events A and B: A B denotes the occurrence of both A and B

Example A = {prime numbers 6) B = {odd numbers 6} A B = {3,5}

S A B

8

Disjoint events

A and B are said to be disjoint if they can not occur simultaneously, i.e. A B = Ø = the empty set

S A B

9

Complementary events

The complement of an event A is all events in the sample space S except for A.

The complement of an event A is denoted by AC

Example A = {even numbers) AC = {odd numbers}

S

A CA

10

Probability

Probability is a set function Pr() which maps events A1, A2,... in the sample space S, to real numbers

The function Pr() can only take values in the interval from 0 to 1, i.e. probabilities are greater or equal than 0, and less or equal than 1

S

0 1

A1 A2

P(A1) P(A2)

11

Kolmogorov basic axioms

1. 0 Pr(A)

2. Pr(S) = 1

3. If A1, A2,... is a sequence of disjoint events we shall then have Pr(A1 A2 ...) = Pr(A1) + Pr(A2) + ...

Everything is based on these axioms in probability calculus

12

Conditional probability

In some situations the probability of A will change if we get information about a related event, say B

We then introduce conditional probabilities, and write: Pr(A|B) = the conditional probability that A will occur

given that B has occurred

Example: Probability of pulling ace of spade is 1/52, but if we have seen a “black” card, the conditional probability is 1/26

13

Independent events

A and B are said to be independent if information about whether B has occurred does not influence the probability that A will occur

Pr(A|B) = Pr(A)

Example: We are both pulling a card and tossing a dice in a composed experiment. The probability of pulling ace of spade (A) is independent of the event getting a six (B)

14

Basic rules for probability calculus

Pr(A B) = Pr(A) + Pr(B) - Pr(A B) Pr(A B) = Pr(A) Pr(B) if A and B are independent Pr(AC)

= Pr(A does not occur) = 1 - Pr(A) Pr(A|B) = Pr(A B) / Pr(B)

15

Example

Let A = {It is Sunday} B = {It is between 6 and 8 pm) A and B are independent but not disjoint We will find Pr(A B) and Pr(A B)

Pr(A B) = Pr(A) Pr(B) =

Pr(A B) = Pr(A)+ Pr(B) - Pr(A B) =

Pr(A|B) =

84

1 =

24

2

7

1

42

9 =

84

1-

24

2+

7

1

1Pr (A B) 1842Pr (B) 7

24

16

Example

Assume we have two redundant shut-down valves, ESDV and PSDV that could be used in an emergency situation Pr(ESDV-failure)=0.01 Pr(PSDV-failure)=0.005

Assuming independent failures give a total failure probability of 0.01 0.005 = 510-5

17

Division of the sample space

A1,A2,…,Ar is said to be a division of the sample space if the union of all Ai’s covers the entire sample space, i.e. A1 A2 … Ar = S and the Ais are pair wise disjoint, Ai Aj = Ø for i j

S

A 1

A 2 A 3

A 4

18

The law of total probability

Let A1,A2,…,Ar represent a division of the sample space S, and let B be an arbitrary event in S, then

1

Pr (B) Pr (B | A ) Pr (A )r

i ii

19

Example

A special component type is ordered from two suppliers A1 and A2

Experience has shown that components from supplier A1 has a defect probability of 1%

components from supplier A2 has a defect probability of 2%

In average 70% of the components are provided by supplier A1

Assume that all components are put on a common stock, and we are not able to trace the supplier for a component in the stock

A component is now fetched from the stock, and we will calculate the defect probability, Pr(B)

1 1 2 21

Pr (B) Pr (B | A ) Pr (A ) Pr (B|A ) Pr (A ) Pr (B|A ) Pr (A )

0.01 0.7 0.02 0.3 1.3%

r

i ii

20

Exercise

Successful evacuation depends on the available evacuation time, A1 = short evacuation time Pr(A1) = 1% A2 = medium evacuation time Pr(A2) = 20% A3 = long evacuation time Pr(A3) = 79%

The probability of successful evacuation (B) is given by: Pr(B| A1) = 50% Pr(B| A2) = 75% Pr(B| A3) = 95%

Find Pr(B) by the law of total probability

21

Random quantities

A random quantity (stochastic variable), is a quantity for which we do not know the value it will take, but We could state statistical properties of the quantity

or make probability statement about it Whereas an event may occur, or not occur (B&W), a

random quantity is related to a magnitude, it may take different values

We use probabilities to describe the likelihood of the different values the random quantity can take Cumulative distribution function (S-curve) Probability density function (histogram)

22

Examples of random quantities

X = Life time of a component (continuous) R = Repair time after a failure (continuous) Z = Number of failures in a period of one year (discrete) M = Number of derailments next year N = Number of delayed trains next month W = Maintenance cost next year

23

Cumulative distribution function (CDF)FX(x) = Pr(X x)

F X(x)

x0

1

24

Exercise

Let X be the life time of a component Use Excel to find Pr(X 150) when FX(x) = 1 - e-(0.01x)

2

x F X (x )

0 0.0010 0.0120 0.0430 0.0940 0.1550 0.2260 0.3070 0.3980 0.4790 0.56

100 0.63110 0.70120 0.76130 0.82140 0.86150 0.89160 0.92170 0.94180 0.96190 0.97200 0.98

0.00

0.10

0.20

0.30

0.40

0.50

0.60

0.70

0.80

0.90

1.00

0 50 100 150 200

25

Probability density function (PDF)

fX(x)

x

)(d

d)( xF

xxf XX

26

PDF probabilities

Pr( ) ( )d ( ) ( )b

X X X

a

a X b f x x F b F a

fX(x)

a bx

27

Expectation

The expectation of a random quantity X, may be interpreted as the long time run average of X, if an infinite amount of observations are available

28

Variance

The variance of a random quantity expresses the variation of X around the expected value in the long run

29

Standard deviation The standard deviation of a random quantity expresses a

typical “distance” from the expected value

30

Parameters describing random quantities

Percentiles, i.e. P1,P10,P50,P90,P99 Most likely value (M) Expected (mean) value () Standard deviation () Variance (Var = 2)

fX(x)

xPx

x%

M m

s

31

Expectation and variance for a sum

Let X1, X2,…, Xn be independent random quantities

We then have

32

Life times

In reliability theory we work with life times The life time, or time to failure, is the time it takes from a

component is installed, until it fails for the first time Life times are non-negative random quantities For life times we introduce the following concepts

R(x) = Pr(X > x) = 1- FX(x) MTTF = Mean Time To Failure = E(X)

33

Statistical view of life times

1

2

3

4

5

6

7

T 1

T 2

T 3

T 4

T 5*

T 6

T 7

t=0 End

34

Distribution classes

Life times are often associated with various distribution classes, e.g. in reliability analysis we often apply the following distribution classes The exponential distribution The Weibull distribution The gamma distribution The normal distribution

35

The exponential distribution

The exponential distribution is a very simple distribution which could be used if no aging affects the component under consideration

Often external or internal shocks dominates the failure causes if the exponential distribution is used

For the exponential distribution we have fX(x) = e-x

FX(x) = 1-e-x

R(x) = e-x

E(X) =1/ Var(X) = 1/2

is a parameter in the distribution (the failure rate)

36

Example

We will obtain the probability that X is greater than it’s expected value. We then have:

Pr(X > E(X)) = R(E(X)) = e-E(X ) = e -1 0.37 i.e., most likely it will not survive the expected life time

37

Example

Assume the life time, X, of a component is exponentially distributed with parameter = 0.01

We will find the probability that the component that has survived 200 hours, will survive another 200 hours

Pr(X > 400 |X > 200) = Pr(X > 400 X > 200)/Pr(X > 200) =Pr(X > 400)/Pr(X > 200) =R(400)/R(200) = e-400/ e-200 = e-200 = Pr(X > 200) Thus, an old component is stochastically as good as a

new component

38

For the Weibull distribution we have R(x) = e-(x)

is a shape parameter, > 1 means aging

MTTF =

Var(X) =

where () is the gamma function

The Gamma function is found in Excel by =EXP(GAMMALN(x))

)1(1 1

)1()1(1 122

2

39

Reparameterization of the Weibull

The Weibull distribution has two parameters: a = shape or aging parameter l = scale parameter

The relation between l and MTTF is MTTF =

In many situations it is easier to work with a and MTTF, rather than a and l

)1(1 1

40

Example

We will find the probability that a component that has survived 200 hours, will survive another 200 hours given that the life time is Weibull distributed with parameter = 2 and = 0.01

Pr(X > 400 |X > 200) = Pr(X > 400 X > 200)/Pr(X > 200) =Pr(X > 400)/Pr(X > 200) =R(400)/R(200) = e-(400)2/ e-(200)2 e-(350)2 < Pr(X > 200)

Thus, an old component is not as good as a new one

41

The hazard rate, z(t)

The hazard rate is the precise term for the so-called bathtub curve, also denoted failure rate funciton:

z(t) = f(t)/R(t) z(t)t Probability of failure in a small time interval (t )

given that the unit has survived up to t.

time, t

Haz

ard

rate

t

42

Example of hazard rates

Exponential distribution z(t) = = constant

Weibull distribution z(t) = ()(t) -1 t -1 = increasing in time t for > 1

Preventive maintenance is often based on the idea of ”taking away” the right hand side of the hazard rate curve