3.1 probability

College of Engineering Engineering Statistics

Department of Dam & Water Resources Lecturer: Goran Adil & Chenar

Chapter 3.1: Basic Probability Concept 1

Introduction to Probability

Learning Objectives

List simple events and sample space.

Basic Counting Rules

Learn the basic rules of probability.

Interpretation and Axioms of Probability

Addition Rule

Conditional Probability

The Law of Total Probability Bayes’s theorem

3.1 Introduction Engineering investigations involving natural phenomena as well as systems devised by humans

exhibit scatter and variability as illustrated in Chapter 2. The resulting uncertainty that the

engineer encounters is a major problem. By using the theory of probability, one can incorporate

this uncertainty into the analysis and thus make rational decisions. The main focus of this chapter

is to define the concept of probability and to discuss some of the associated axioms and basic

properties

Probability can be defined as the chance of an event occurring. It can be used to quantify what

the “odds” are that a specific event will occur. Some examples of how probability is used every

day would be weather forecasting, “75% chance of snow” or for setting insurance rates.

The concept of probability plays an important role in our daily lives. Assume you have an

opportunity to invest some money in a software company. Suppose you know that the company’s

records indicate that in the past five years, its profits have been consistently decreasing. Would

you still invest your money in it? Do you think the chances are good for the company in the

future?




3.2 Sample space and events

The Random experiments

An experiment which produces different results, even though it is repeated many times in the

same manner, under essentially similar conditions, is called a random experiment.

A set conditions under which behaviour of some variables are observed.

- Example: Gauge station, testing soil/concrete samples, flipping coins, gambling, etc.

Definition: Sample space.

The sample space, denoted by S, is the collection of all possible events arising from a conceptual

experiment or from an operation that involves chance.

Example 3.1

Sample space for tossing a coin, S = ( , )

Sample space for tossing a die, S= ( , , , , , )

The sample space for tossing two coins once (or tossing a coin twice) will contain four

possible outcomes denoted by

S =

HINT In this example, clearly, S is the Cartesian product A A,

Then The sample space S for the random experiment of throwing two six-sided dice can be

described by the Cartesian product A A, where

S (1 dice) = {1, 2, 3, 4, 5,6}. In other words, S = A A hence,

S =




Example 3.2 A contractor must select a strategy for a construction job. Two independent operations, I and II,

must be performed in succession. Each operation may require 4,5, or 6 days to complete.. M4 is

the event that operation I requires 4 days, N4 that II requires 4 days, etc. Construct a graphical

representations of the utilities-demand sample space.

Example 3.3 consider the design of an underground utilities system for a building sites. The sites have not yet

been leased, and so the nature of occupancy of each is not known. If the engineer provides water

and power capacities in excess of the demand actually encountered. Consider any particular site

and assume that the electric power required by the occupant will be either 5 or 10 units, while the

water capacity demanded will be either 1 or 2 units.

Construct a graphical representations of the utilities-demand sample space.




Events Event is a possible outcome or set of possible outcomes of the experiment. Typically denoted by

a capital letter near the beginning of the alphabet: A, B etc. In other words, any subset of a

sample space S of a random experiment, is called an event.

SIMPLE & COMPOUND EVENTS:

An event that contains exactly one sample point (has only one outcome is an), is defined as a

simple (Elementary) event. A compound event is event which has more than one outcome

Ex.. The occurrence of a 6 when a die is thrown, is a simple event, while the occurrence of a

sum of 10 with a pair of dice, is a compound event, as it can be decomposed into three simple

events (4, 6), (5, 5) and (6, 4).

Types of Event

Null Set

An event which does not contain any outcome is a null event (impossible event). It is denoted by

Φ.

Example: Set of month with 32 days , Getting in 7 in rolling a single dice

Sure Event

An event which contains all the outcomes is equal to the sample and it is called sure event or

certain event.

- A Certain (sure) Event is an event that is sure to occur (probability = 1)

COMPLEMENTARY EVENT:

The event “not-A” is denoted by A or Ac and called the negation (or complementary event) of

A.

EXAMPLE: If we toss a coin once, then the complement of “heads” is “tails”.

If we toss a coin four times, then the complement of “at least one head” is “no heads”.

MUTUALLY (Dis-joint) EXCLUSIVE EVENTS:

Two events A and B of a single experiment are said to be mutually exclusive or disjoint if and only

if they cannot both occur at the same time i.e. they have no points in common.

Ex.When we toss a coin, we get either a head or a tail,

but not both at the same time.

The two events head and tail are therefore mutually exclusive.




Independent event If the Event (E1) occurred and did not affect the occurrence of event (E2), they are called

Independent Events.

Ex: If we toss two coins, getting heads with the first coin will not affect the probability of getting

heads with the second.

Non independent event The occurrence of event (E1) is affecting the occurrence of event (E2).

Ex: In a box containing white and black balls, the pulling of two balls respectively without

returning the first ball that means the first pulling is affecting the second pulling.

Possible cases They are all different cases that could be resulted from the experiment.

Ex: Toss of a coin results in two possible cases (H or T).

Impossible Event – an event that has no chance of occurring (probability = 0)

Favorable cases They are the cases where required events occurred. They are the success cases.

Ex: Toss of a dice, if the required event is the even numbers, then the cases are (2),(4),(6) which

called favourable cases.

EQUALLY LIKELY EVENTS:

Two events A and B are said to be equally likely, when one event is as likely to occur as the other. In other

words, each event should occur in equal number in repeated trials.

Ex. When a fair coin is tossed, the head is as likely to appear as the tail, and the proportion of times each

side is expected to appear is 1/2.




Tree diagram Tree diagrams allow us to see all the possible outcomes of an event and calculate their

probability. Each branch in a tree diagram represents a possible outcome.

Example 3.4 The sequence of construction of a structure involves two phases. Initially, the foundation is built,

then work commences on the superstructure. The completion of the foundation can take 4 or 5

months, which are equally likely to be needed. The superstructure requires 5, 6, or 7 months to

be completed, with equal likelihood for each period. The time of completion of the

superstructure is independent of that taken to complete the foundation. List the possible

combinations of times for the completion of the project and determine the associated

probabilities.

Vein Diagram Vein Diagram is a graphical representation that is useful for illustrating logical relations among

events is the Venn diagram. The sample space S is represented as consisting of all the outcomes

in a large rectangle,




Set A set is any well-defined collection or list of distinct objects, e.g. a group of students, the books

in a library, the integers between 1 and 100, all human beings on the earth, etc

SUBSETS:

A set that consists of some elements of another set, is called a subset of that set.

For example, if B is a subset of A, then every member of set B is also a member of set A.

OPERATIONS ON SETS:

Let the sets A and B be the subsets of some universal set S. Then these sets may be combined

and operated on in various ways to form new sets which are also subsets of S.

The basic operations are union, intersection, difference and complementation.

UNION OF SETS:

The union or sum of two sets A and B, denoted by A B, and read as “A union B”, means

the set of all elements that belong to at least one of the sets A and B, that is

By means of a Venn Diagram, A B is shown by the shaded area as below:

INTERSECTION OF SETS:

The intersection of two sets A and B, denoted by A B, and read as “A

intersection B”, means that the set of all elements that belong to both A and B; that is

A B Diagrammatically, A B is shown by the shaded area as below:

Example:

Let A = {1, 2, 3, 4} and B = {3, 4, 5, 6}

Then A B = {1, 2, 3, 4, 5, 6}

Then A B = {3, 4}




COMPLEMENTATION:

The set of all those elements of S which do not belong to A, is called the complement of A and is

denoted byA or by Ac.

In symbols:

A = {x | x S and s A}

The complement of A is shown by the shaded portion in the following Venn diagram.




Let A, B and C be any subsets of the universal set S. Then, we have:

Commutative laws:

A B = B A and

A B = B A

Associative laws:

(A B) C = A (B C)

And

(A B) C = A (B C)

Distributive laws

A (B C) = (A B) (A C)

and

A (B C) = (A B) (A C)




3.3 The Random Experiment

Sampling Techniques:

With replacement Each item in the sample space is replaced before the next draw

Example 3.5 Suppose a box contains three balls, one red, one blue, and one white. One ball is selected, its

color is observed, and then the ball is placed back in the box. The balls are scrambled, and again,

a ball is selected and its color is observed. What is the sample space of the experiment?

It is probably best if we draw a tree diagram to illustrate all the possible selections.

Without replacement Samples are drawn without replacement in the sample space.

Example 3.6 Consider the same experiment as in the last example. This time we will draw one ball and record

its color, but we will not place it back into the box. We will then select another ball from the box

and record its color. What is the sample space in this case?

Solution: The tree diagram below illustrates this case:




3.4 Fundamental Counting Principle Sometimes, instead of writing out all the outcomes for a sample space, we instead consider the

counts of the number of outcomes for analysis

1- Combinations

2- Permutations

Factorial n!, read n factorial, is the product of all the counting numbers less than or equal to n.

7! = 7 6 5 4 3 2 1

6! = 6 5 4 3 2 1

1! = 1

Also, we define 0! = 1.

Multiplication Rule

Example 3.7 A certain make of automobile is available in any of three colors: red, blue, or green, and comes

with either a large or small engine. In how many ways can a buyer choose a car?

There are three choices of color and two choices of engine. A complete list of choices is written

in the following 3×2 table.

The total number of choices is

If an operation can be performed in n1 -ways, and if for each of these ways a second operation

can be performed in n2 ways, then the two operations can be performed together in n1× n2 ways.




Permutation A Permutation is an arrangement of items in a particular order.

Sometimes we are concerned with how many ways a group of objects can be arranged

1) How many ways to arrange books on a shelf

2) How many ways a group of people can stand in line

Permutation Formula (without replacement)

!

( )!

n

r

nP

n r

Requirements:

1. There are n different items available. (This rule does not apply if some of the items are

identical to others.)

2. We select r of the n items (without replacement).

3. We consider rearrangements of the same items to be different sequences. (The permutation of

ABC is different from CBA and is counted separately.)

Permutation Formula (with replacement)

n r

rP n

Combination Sometimes, we are only concerned with Selecting a group and not the order in which

they are selected.

A combination gives the number of ways to select of r objects from a group of size n.

You have n object

You want a group of r object

You DON’T CARE what order they are selected in

! = where 0 .

!( )!

nC r n

n r r n r

Combinations are also denoted nCr Read “n choose r”

Notice, ORDER MATTERS




Example 3.8 Suppose the question is matching: there are 6 questions and 10 possible choices. Now, how

many ways can you match?

10

6

10! 10! 10 9 8 7 6 5 4!151200

(10 6)! (4)! (4)!

n

rP Ways

or

Example 3.9 The number of ways to arrange the letters ABC

3! 3!3 2 1 6

(3 3)! (1)!

n

rP

WAYS

OR Similarly:

ABC ACB BCA BAC CAB CBA

Example 3.10 Suppose you have 4 letters A, B, C, D. In how many ways you can SELECT 3 letters ?

ABC ABD ACD BCD

OR

! 4! = 4

!( )! 3!(4 3)!

nC Ways

n r r n r

Example 3.11 Suppose you have 4 letters A, B, C, D. In how many ways you can ARANGE 3 letters ?

ABC ACB ABD ADB ACD ADC BAC BCA BAD BDA BCD BDC

CAB CBA CAD CDA CBD CDB DCB DBC DAC DCA DAB DBA

OR

4!

(4 3)!

n

rP

=24 WAYS




Probability Concept

Probability is a number associated to events, the number denoting the ’chance’ of that event

occurring. Words like “probably,” “likely,” and “chances” convey similar ideas. They convey

some uncertainty about the happening of an event. In Statistics, a numerical statement about the

uncertainty is made using probability with reference to the conditions under such a statement is

true

An example of problems that requires probability theory: A town is protected from floods by a reservoir dam that is designed for a 50-year flood; that is,

the probability that the reservoir will overflow in a year is 1/50 or 0.02. The town and reservoir

are located in an active seismic region; annually, the probability of occurrence of a destructive

earthquake is 5%. During such an earthquake, it is 20% probable that the dam will be damaged,

thus causing the reservoir water to flood the town. Assuming that the occurrences of natural

floods and earthquakes are statistically independent.

(a) What is the probability of an earthquake-induced flood in a year?

(b) What is the probability that the town is free from flooding in any one year.

(c) If the occurrence of an earthquake is assumed in a given year, what is the probability

that the town will be flooded that year?




Basic Rules for Computing Probability

Rule 1: Relative Frequency Approximation of Probability

Conduct (or observe) a procedure, and count the number of times event A actually occurs. Based

on these actual results, P (A) is approximated as follows:

Number of times A occured( )

#of times procedure was repeated

nP A

N

Rule 2: Classical Approach to Probability (Requires Equally Likely Outcomes)

Assume that a given procedure has n different simple events and that each of those simple events

has an equal chance of occurring. If event A can occur in s of these n ways, then

Number of ways A can occur( )

# of different simple events

nP A

N

Rule 3: Subjective Probability P(A), the probability of event A, is estimated by using knowledge of the relevant circumstances.

Note

Denote events by roman letters (e.g., A, B , etc)

Denote probability of an event as P (A)

Axioms of Probability

1. Axiom 1 ( ) 0 for any event P A A

2. Axiom 2 ( ) 1 P S

3. The probability of an impossible event is 0.

4. The probability of an event that is certain to occur is 1.

If all Ai’s are mutually exclusive, then

1 2

1

Axiom 3 ( ... ) ( )k

k i

i

P A A A P A

1 2

1

( ...) ( )i

i

P A A P A

(finite set)

(infinite set)




Properties of Probability

1- Complement (non-Probability) Complement of an event is that the event did not occur.

1 ( ).P A P A cA A A

2- Complements: The Probability of “At Least One” “At least one” is equivalent to “one or more.”

The complement of getting at least one item of a particular type is that you get no items of

that type.

To find the probability of at least one of something, calculate the probability of none, then

subtract that result from 1. That is,

1 (None).P at least one P

3- Addition Rule If A and B are two events, then

P (A ∪ B) = P (A) + P (B) − P (A ∩ B).

If they are mutually exclusive (disjoint), then

Events A and B are disjoint (or mutually exclusive) if they cannot both occur together

P (A ∪ B) = P (A) + P (B).




4- Conditional Probability

The probability of an event B occurring when it is known that some event A has occurred is

called a conditional probability and is denoted by P (B|A). The symbol P (B|A) is usually read

“the probability that B occurs given that A occurs” or simple the probability of B, given A.

For any two events A and B with P (A) > 0, the conditional probability of B given that A has

occurred is:

|

P A BP A B

P B

P (B|A): pronounced "the probability of B given A.”

Which can be written:

|P A B P B P A B = Multiplication Rule

Independence Independent Events

Two event A and B are independent events if ( | ) ( ).P A B P A

Otherwise A and B are dependent.

Events A and B are independent events if and only if

( ) ( )P A B P A P B

P (A B) = P (A) + P (B) − P (A ∪ B).




Example 3.11

Flood occurrence. Consider the floods that exceed the previously established design flood in the

outlet reach of the Bisagno River at Genoa, Italy, observed from 1931 to 1995. Records indicate

that six floods occurred in the period, namely, in 1945, 1951 (twice), 1953, 1970, and 1992. Let

N Let N denote the number of flood occurrences per year.

- Find the probability that at least one flood occurs in any years?

- Find the probability that at least two flood occurs in any year?

- Find that probability that no flood occurs in any year ?

- Given that flooding had occurred, find the probability of two flood per year ?




Example 3.12

Consider the n = 40 paired data of densities and compressive strengths of concrete given in Table

below.

Density

kg/m3

Strength

N/mm3

Density

kg/m3

Strength

N/mm3

Density

kg/m3

Strength

N/mm3

Density

kg/m3

Strength

N/mm3

Density

kg/m3

Strength

N/mm3

2437 60.5 2441 61.9 2436 59.6 2448 59.0 2446 60.9

2437 60.9 2456 67.2 2450 60.5 2445 63.3 2445 60.0

2425 59.8 2444 64.9 2454 59.8 2436 52.5 2415 50.7

2427 53.4 2447 63.4 2449 56.7 2469 54.6 2411 58.8

2428 56.9 2433 60.5 2441 57.9 2455 56.3 2427 54.4

2448 67.3 2429 68.1 2457 60.2 2473 64.9 2458 61.1

2456 68.9 2435 68.3 2447 55.8 2488 69.5 2472 61.5

2436 49.9 2471 65.7 2436 53.2 2454 58.9 2435 57.8

For The following defined events:

A ≡ {2440 < λc < 2460 kg/m3} and B ≡ {55 < ηc < 65 N/mm2},

where λc denotes the density of a concrete cube under test, measured in kg/m3, and ηc denotes the

compressive strength of that cube, measured in N/mm2.

Find

a)

PA, PAC PB PBC P (A ∩ B).

P (A ∪ B)

b) The probability that a concrete cube with density from 2440 to 2460 kg/m3 yields a value of

compressive strength in the range 55–65 N/mm2 is




The Law of Total Probability If the events A1, A2,…, An be mutually exclusive and exhaustive events. Then for any other

event B,

1

( | ) ( )n

i i

i

P B P B A P A

Baye’s Theorem Let A1, A2, …, An be a collection of k mutually exclusive and exhaustive events with P(Ai) > 0 for

i = 1, 2,…, n. Then for any other event B for which P(B) > 0,

/ A|

P B i P AiP Ai B

P B =

1

||

|

j j

j n

i i

i

P A P B AP A B

P A P B A

1,2...,j k

Example 3.13

Aggregates used for highway construction are produced at three plants with daily production

volumes of 500, 1000,and 2000 tons. Past experience indicates that the fractions of deleterious

materials produced at the three plants are, respectively, 0.005, 0.008, and 0.010. If a sample of

aggregate is selected at random from a day's total production and found to be deleterious, which

plant is likely to have produced the sample?

Tutorials




TUTORIAL 1

A firm is deciding to build two new plants, one in the east and one in the west. Four eastern cities

(A, B, C, D) and two western cities (E, F). Construct tree diagram.

- Draw a tree diagram to represent the drawing two marbles from a bag containing blue,

green, and red marbles and determine the probability of getting at two blue marbles

- Suppose you have 5 multiple-choice problems tomorrow, each with 4 choices. How

many different ways can you answer these problems?

- In how many ways can a President, Vice President, Secretary, and Treasurer be elected

for a club having fifty members?

- From a club of 24 members, a President, Vice President, Secretary, Treasurer and

Historian are to be elected. In how many ways can the offices be filled?

- What are the ways of arrangements that you can form out of the word “STATISTICS”?

- An inspector randomly selects 2 of 5 parts for inspection. In a group of 5 parts, how

many combinations of 2 parts can be selected?

- Iowa randomly selects 6 integers from a group of 47 to determine the weekly winner.

What are your odds of winning if you purchased one ticket?

- At a company with 35 engineers, the boss will be choosing 5 to go to a conference. How

many different groups of 5 members are there to choose from?

- For a `fair' die with equally likely outcomes, what is the probability of rolling an even?

- A coin is tossed twice. What is the probability that at least one head occurs?

- Consider the experiment of tossing a coin ten times. What is the probability that we will

observe at least one head?

- What is the probability of getting a total of 7 or 11 when pair of fair dice is tossed?

- a fair dice are rolled. What is the probability of getting a sum less than 7 or a sum equal

to 10?




- If you know that 84.2% of the people arrested in the mid 1990’s were males, 18.3% of

those arrested were under the age of 18, and 14.1% were males under the age of 18, what

is the probability that a person selected at random from all those arrested is either male or

under the age of 18?

- A town has two fire engines operating independently. The probability that a specific

engine is available when needed is 0.96. (a) What is the probability that neither is

available when needed? (b) What is the probability that a fire engine is available when

needed?

- Roll a dice. What is the chance that you would get a 6, given that you’ve gotten an even

number?

- A college class has 42 students of which 17 are male and 25 are female. Suppose the

teacher selects two students at random from the class. Assume that the first student who

is selected is not returned to the class population. What is the probability that the first

student selected is female and the second is male?

- In a recent election, 35% of the voters were democrats and 65% were not. Of the

democrats, 75% voted for candidate Z, and of the non-Democrats, 15% voted for

candidate Z. Define the following events:

A = voter is Democrat, B = voted for candidate Z

1. Find P(B|A); P(B|Ac)

2. Find P(A ∩ B) and explain in words what this represents.

3. Find P(Ac ∩ B) and explain in words what this represents

- The king comes from a family of 2 children. What is the probability that the other child is

his sister? ans=2/3

- A couple has 2 children. What is the probability that both are girls if the older of the two

is a girl? ans= ½

- If P(C)= 0.65, P(D)= 0.4, and P(C D )=0.26, are the event C and D independent ?

- A small town has one fire engine and one ambulance available for emergencies. The

probability that the fire engine is available when needed is 0.98, and the probability that

the ambulance is available when called is 0.92. In the event of an injury resulting from a

burning building, find the probability that both the ambulance and the fire engine will be

available, assuming they operate independently.




- In 1970, 11% of Americans completed four years of college; 43% of them were women.

In 1990, 22% of Americans completed four years of college; 53% of them were women

(Time, Jan. 19, 1996).

(a) Given that a person completed four years of college in 1970, what is the probability that the

person was a woman?

(b) What is the probability that a woman finished four years of college in 1990?

(c) What is the probability that a man had not finished college in 1990?

- A town has two fire engines operating independently. The probability that a specific

engine is available when needed is 0.96.(a) What is the probability that neither is

available when needed?(b) What is the probability that a fire engine is available when

needed?

TUTORIAL 2

1. If S = {0,1,2,3,4,5,6,7,8,9} and A ={0,2,4,6,8}, B={1,3,5,7,9}, C={2,3,4,5}, and D={1,6,7},

list the elements of the sets corresponding to the following events:

a) A∪C;

b) A∩B; c) (S∩C)c

d) A∩C∩D

e) Cc

2. Let A, B, and C be events relative to the sample space S. Using Venn diagrams, shade the

areas representing the following events:

a) (A∩B)c

b) (A∪B)c

TUTORIAL 3

Reservoir inflows. A reservoir impounds water from a stream X and receives water Y deviated

via a tunnel from an adjoining catchment. The annual inflow from source X can be approximated

to 1 or 2 or 3 units of 106 m3, and that from source Y is 2 or 3 or 4 units of 106 m3. On

appropriate Venn diagrams show the following events:

(a) A ≡ {source X is less than 3 units} (b) B ≡ {source Y is more than 2 units} (c) A + B. (d) AB.




TUTORIAL 4

The sequence of construction of a structure involves two phases. Initially, the foundation is built,

then work commences on the superstructure. The completion of the foundation can take 4 or 5

months, which are equally likely to be needed. The superstructure requires 5, 6, or 7 months to

be completed, with equal likelihood for each period. The time of completion of the

superstructure is independent of that taken to complete the foundation. List the possible

combinations of times for the completion of the project and determine the associated

probabilities.

TUTORIAL 5

Dam spillway. An engineer is designing a spillway for a dam. The evaluation of maximum flow

data is based on a short period of recordkeeping. The critical flow rates and their probabilities are

estimated from, A, discharge measurements, B, rainfall observations, and C, combination of flow

discharge and rainfall data, as follows:

Event A from flow data: 8,000 to 12,000 m3/s, Pr[A] = 0.5.

Event B from rainfall data: 10,000 to 15,000 m3/s, Pr[B] = 0.6.

Event C = A + B: 8,000 to 15,000 m3/s, Pr[C] = 0.9.

(a) Sketch the foregoing events.

(b) Show on the sketch AB, AC, and Ac + Bc.

(c) Determine the probabilities Pr[AB] and Pr[Ac + Bc].

(d) Determine the conditional probabilities Pr[A|B] and Pr[B|A].

TUTORIAL 6

Irrigation water supply. A dam is designed to supply water to three separate irrigation schemes,

I1, I2, and I3. The demand for the first scheme I1 is 0 or 1 or 2 m3/s, whereas that for I2 and I3 is

0 or 2 or 4 m3/s in each case.

(a) Sketch the sample space for I1, I2, and I3 separately, and for I1, I2, and I3

jointly.

(b) Show the following events:




A ≡ {I1 > 1 m3/s}; B ≡ {I2 ≥ 2 m3/s}; C ≡ {I3 < 4 m3/s};

Ac;AB; A + B; (A + B)c; ABc; AC; AcC; BcC; BcCc; (where feasible).

a) Assuming that the demands from the three schemes are independent of each other, and

that all possible demands are equally likely to occur, find the probability that the total

water demand exceeds 5 m3/s.

TUTORIAL 7

Hydropower. Run-of-river hydroelectrical plants convert the natural potential energy of surface

water in a stream into electrical energy. The plant capacities depend on natural river flow, which

generally varies during the year according to season and precipitation regime. Assume that the

design flow of a given power station, say, QD, is the natural flow, which is exceeded during 274

days in a year on average. At other times, when the river flow is lower than the design flow, the

plant is nevertheless capable of producing some power if the flow is not lower than Q0.

Moreover, during floods it is not possible to convey water to the plant due to sedimentation,

which occurs when the natural river flow Q exceeds Q1.

(a) If Pr[Q < Q0] = 0.1 and Pr[Q > Q1] = 0.05, for how many days in a year will the plant be

incapable of supplying electric energy?

(b) What is the probability that the plant works at full capacity?

(c) What is the probability that the plant fulfills its minimum target? Note that

Q0 < QD < Q1.

TUTORIAL 8

Industrial park utilities. Consider the design requirements of water supply and wastewater

removal systems in a new industrial park, which consists of five independent buildings. Assume

that the water demand S of each of the five industrial buildings can be 10 or 15 units, whereas the

required wastewater removal capacity R can be 8, 10, or 15 units. After some interviews with

potential clients, the designer has estimated that the combined requirements of the two systems

are likely to occur with the following probabilities at the i-th site:

R=15 R=10 R=8




S=10 00 0.25 0.15

S=15 0.2 0.35 0.2

Independence can also be assumed among the requirements of different buildings.

(a) What is the probability that the total water demand exceeds 60 units?

(b) What is the probability that the total wastewater removal capacity exceeds 50

units?

TUTORIAL 9

Two power generation units A and B operates in parallel to supply the power requirements of a

small city. The demand for water is subjected to considerable fluctuation, and it is known that

each unit has a capacity of supplying the city’s full requirement 75% of the time in case the other

unit failed. The probability of failure of each unit is 0.10, where the probability that both unit

will fails is 0.02.

If there is a failure in the power generation, what is the probability that the city have its supply of

full water. ?

TUTORIAL 10

Pumping station. Two pumps operate in parallel to provide water supply of a village located in

a recreational area. Water demand is subject to considerable weekly and seasonal fluctuations.

Each unit has a capacity so that it can supply the demand 80% of the time in case the other unit

fails. The probability of failure of each unit is 10%, whereas the probability that both units fail is

3%. What is the probability that the village demand will be satisfied?

TUTORIAL 11

A town is protected from floods by a reservoir dam that is designed for a 50-year flood; that is

,the probability that the reservoir will overflow in a year is 1/50 or 0.02. The town and reservoir

are located in an active seismic region; annually, the probability of occurrence of a destructive

earthquake is 5%. During such an earthquake, it is 20% probable that the dam will be damaged,

thus causing the reservoir water to flood the town. Assuming that the occurrences of natural

floods and earthquakes are statistically independent.

(a) What is the probability of an earthquake-induced flood in a year?

(b) What is the probability that the town is free from flooding in any one year.




(c) If the occurrence of an earthquake is assumed in a given year, what is the probability

that the town will be flooded that year?

TUTORIAL 12

Construction scheduling. Consider the sequential construction scheme of Problem 2.3, and

assume that both the foundation and the superstructure can be completed at three different rates,

say, a, b, or c. These rates modify the probability of completion of each phase of construction as

shown in the table given here. Also, monthly costs vary for the different rates.

In addition, if the construction is not completed in 11 months, the contractor must pay a penalty

of $300,000 per month.

(a) Compute the expected cost of foundation performed at rate a as the summation

for all times of completion of the product between the total cost (the product of the number of

required months and the cost per month) and probability.

(b) Compute all expected costs.

(c) Compute the total expected penalty for each possible strategy of completion of the whole

structure.

(d) Determine the best strategy by minimizing the sum of total expected cost and penalty

TUTORIAL 13

Analysis of reservoir lifetime. A reservoir is designed for an area with high erosional rates. The

engineer is interested in determining the lifetime of the reservoir, which can come to an end

either because the impounding dam can be destroyed by a flood exceeding the spillway capacity

or because excessive sedimentation results in a severe loss in reservoir capacity. It is necessary to

determine the probability that the structure will come to an end of its useful life in each of the

years after construction. One can assume a constant probability q that in any year a flow




exceeding the spillway capacity can occur, and an exponentially increasing probability pi that

reservoir sedimentation can occur in the ith year after construction, given that no significant

sedimentation has occurred prior to the ith year, that is,

Denote by An the event associated with a destructive flood occurring in the nth

year after construction and by Bn that associated with excessive sedimentation.

(a) What is the probability that the system will survive for n years, that is

(b) What is the probability that the system will come to an end in the nth year,

where Sn denotes survival up to the nth year,

Pr[(An + Bn)|Sn−1] Pr[Sn−1]?

(c) Compute the foregoing probabilities for q = 0.01, β = 0.002, and n = 25.

TUTORIAL 14

Dam failure. Two natural events can result in the failure of a dam in an earthquake-prone area.

Firstly, a very high flood, exceeding the design capability of its spillway, say, event A, may

destroy it. Secondly, a destructive earthquake can cause a structural collapse, say, event B.

Hydrological and seismological consultants estimate that the probability measures characterizing

flood exceedance and earthquake occurrence on a yearly basis are Pr[A] = 0.02 and Pr[B] = 0.01

respectively. The occurrence of one or both events can result in the failure of the dam.

Find the probability of the dam failure.

TUTORIAL 15

Highway pavement. Before any 250-m length of a pavement is accepted by the State Highway

Department, the thickness of a 30 cm is monitored by an ultrasonics instrument to verify

compliance to specification. Each section is rejected if the measured thickness is less than 10 cm;

otherwise, the entire section is accepted. From past experience, the State Highway engineer

knows that 85% of all sections constructed by the contractor comply with specifications.

However, the reliability of ultrasonic thickness testing is only 75%, so that there is a 25% chance




of erroneous conclusions based on the determination of thickness with ultrasonics.

(a) What is the probability that a poorly constructed section is accepted on the basis of the

ultrasonic test?

(b) What is the probability that if a section is well constructed, it will be rejected on the basis of

the ultrasonic test?

TUTORIAL 16

An oil company is bidding for the rights to drill a well in field A and a well in field B. The

probability it will drill a well in field A is 40%. If it does, the probability the well will be

successful is 45%. The probability it will drill a well in field B is 30%. If it does, the probability

the well will be successful is 55%. Calculate each of the following probabilities:

a) probability of a successful well in field A,

b) probability of a successful well in field B,

c) probability of both a successful well in field A and a successful well in field B,

d) probability of at least one successful well in the two fields together,

e) probability of no successful well in field A,

f) probability of no successful well in field B,

g) probability of no successful well in the two fields together (calculate by two methods),

h) probability of exactly one successful well in the two fields together.

Show a check involving the probability calculated in part h.

TUTORIAL 17

An engineer is designing a large culvert to carry the runoff from two separate areas. The quantity

of water from area A may be 0, 10, 20, 30 cfs and that from B may be 0, 20, 40, 60 cfs. Sketch

the sample spaces for A and B jointly and for A and B separately. Define the following events

graphically on the sketches.




TUTORIAL 18

The foundation of a wall can fail either by excessive settlement or from bearing capacity. The

respective failures are represented by events A and B, with probabilities Pr[A] = 0.005, and Pr[B]

= 0.002. The probability of failure in bearing capacity given that the foundation displays

excessive settlement is Pr[B|A] = 0.2, say.

Find

- The probability of failure of the wall foundation ?

- The probability that there is excessive settlement in the foundation but there is no failure

in bearing capacity

- The probability that the foundation has excessive settlement given that it fails in bearing

capacity is obtained from Eq. (2.2.11) as follows:

TUTORIAL 19

Water distribution. Consider a pipeline for the distribution of a water supply of an urban area

of 200 km2. The city plan is approximately rectangular with dimensions of 10 by 20 km, and it is

uniformly covered by the network shown in Figure below. Pressures and flow rates are uniform

throughout the whole network, so that losses are equally likely to occur within it.




Fig. 2.2.3 Pipeline network for urban water supply.

Assume that the probability of a loss in a given subarea is proportional to the area. Therefore,

if a loss occurs in the pipe network,

For the defined events

A ≡ {a severe water loss occurs in location u ≡ (u1, u2)

where 0 < u1 ≤ 6 km, 0 < u2 ≤ 3 km} and

B ≡ {a severe water loss occurs in location v ≡ (v1, v2)

where 4 < v1 ≤ 12 km, 2 < v2 ≤ 6 km}.

Find

a)

PA, PAC PB PBC P (A ∩ B).

P (A ∪ B)

If a loss occurs in the area affected by event B, what is the probability of event A?”

TUTORIAL 20

A question of the acceptability of an existing concrete culvert to carry an anticipated flow has

arisen. Records are sketchy, and the engineer assigns estimates of annual maximum flow rates

and their likelihoods of occurrence (assuming that a maximum of 12 cfs is possible) as follows:




(a) Construct the sample space. Indicate events A, B, C, A ∩ C, A ∩ B, and Ac ∩Bc on the

sample space.

TUTORIAL 21

In the study of a storage-dam design, it is assumed that quantities can be measured sufficiently

accurately in units of ¼ of the dam’s capacity. It is known from past studies that at the beginning

of the first (fiscal) year the dam will be either full, ¾ full, ½ full, or ¼ full, with probabilities ⅓,

⅓, , and , respectively. During each year water is released. The amount released is ½ the capacity

if at least this much is available; it is all that remains if this is less than ½ the capacity. After

release, the inflow from the surrounding watershed is obtained. It is either ½ or ¼ of the dam’s

capacity with probabilities ⅔ and ⅓, respectively. Inflow causing a total in excess of the capacity

is spilled. Assuming independence of annual inflows, what is the probability distribution of the

total amount of water at the beginning of the third year?

TUTORIAL 22

A large dam is being planned, and the engineer is interested in the source of fine aggregate for

the concrete. A likely source near the site is rather difficult to survey accurately. From surface

indications and a single test pit, the engineer believes that the magnitude of the source has the

possible descriptions: 50 percent of adequate; adequate; or 150 percent of possible demand. He

assigns the following probabilities of these states.




Prior to ordering a second test pit, the engineer decides that the various likelihoods of the

sample’s possible indications (Z1 Z2, Z3) depend upon the (unknown) true state as follows:

What are the probabilities of observing the various events Z1, Z2, and Z3?

The second test pit is dug and the source appears adequate from this pit. Compute the posterior

probabilities of state. If another test pit gives the same result, calculate the second set of posterior

state probabilities. Compare prior and posterior state probabilities.

TUTORIAL 23

A quality-control plan for the concrete in a nuclear reactor containment vessel calls for casting 6

cylinders for each batch of 10 yd3 poured and testing them as follows:

1 at 7 days 1 at 14 days 2 at 28 days

2 more at 28 days if any of first four cylinders is “inadequate”

The required strength is a function of age.

If the cylinder to be tested is chosen at random from those remaining (i. e., with equal

likelihoods):

(a) What is the probability that all six will be tested if in fact one inadequate cylinder exists in

the six?




b) If the batch will be “rejected” if two or more inadequate cylinders are found, what is the

likelihood that it will not be rejected given that exactly two are in fact inadequate? (Rejection

will lead to more expensive coring and testing of concrete in place.)

(c) A “satisfactory” concrete batch gives rise to an inadequate cylinder with probability p = 0.1.

(This value is consistent with present recommended practice.) What is the probability that there

will be one or more inadequate cylinders in the six when the batch is “satisfactory”? (Assume

independence of the quality of the individual cylinders.)

(d) Given that the batch is satisfactory (p = 0.1), what is the probability that the batch will be

rejected? What is the probability that an unsatisfactory batch (in particular, say, p = 0.3) will not

be rejected? Clearly a quality control plan wants to keep both these probabilities low, while also

keeping the cost of testing small.

TUTORIAL 24

A major city transports water from its storage reservoir to the city via three large tunnels. During

an arbitrary summer week there is a probability q that the reservoir level will be low. Owing to

the occasional call to repair a tunnel or its control valves, etc., there are probabilities pi(i = 1, 2,

3) that tunnel i will be out of service during any particular week. These calls to repair particular

tunnels are independent of each other and of the reservoir level. The “safety performance” of the

system (in terms of its potential ability to meet heavy emergency fire demands) in any week will

be satisfactory if the reservoir level is high and if all tunnels are functioning; the performance

will be poor if more than one tunnel is out of service or if the reservoir is low and any tunnel is

out of service; the performance will be marginal otherwise.

(a) Define the events of interest. In particular, what events are associated with marginal

performance?

(b) What is the probability that exactly one tunnel fails?

(c) What is the probability of marginal performance?

(d) What is the probability that any particular week of marginal performance will be caused by a

low reservoir level rather than by a tunnel being out of service?




TUTORIAL 25

A gravity retaining wall may fail either by sliding (A) or overturning (B)or both)

Assume:

- Probability of failure by sliding is twice as likely as that by overturning; that is,

P(A) = 2P(B).

( ii ) Probability that the wall also fairs by sliding, given that it has failed by, overturning,

P(A ) /(B) = 0.8

(iii) The probability of failure of wall =0.001

(a) Determine he probability that sliding will occur. Ans. O.00091.

b)lf the wall fails, what is the probability that only sliding has occurred? Ans. 0.546

Backfilled Soil

3.1 probability

Engineering

Transcript of 3.1 probability