3.2 probablity

College of Engineering Engineering Statistics
Department of Dam & Water Resources Lecturer: Goran Adil & Chenar
Introduction to Probability Chapter 3-2 60
Learning Objectives
Operation on Sets
Interpretation and Axioms of Probability
Addition Rules
Conditional Probability
Multiplication and Total Probability Rules
Probability
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
The package says the probability that the bulb I planted will grow is 0.90 or
90%."
There's a high probability that my car will break-down this month."
Probabilities for a random experiment are often assigned on the basis of a
reasonable model of the system under study.
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:
Introduction to Probability
#of times A occured
( )
#of times procedure was repeated
n
P 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
Rule 3: Subjective Probabilities
P(A), the probability of event A, is estimated by using knowledge of the
relevant circumstances.
Note
Elementary events are equally likely
Denote events by roman letters (e.g., A, B , etc)
Denote probability of an event as P (A)
Example 1:
For a `fair' die with equally likely outcomes, what is the probability of rolling
an even?
Example 2:
A coin is tossed twice. What is the probability that at least one head occurs?
Example 3:
A vehicle arriving at an intersection can turn left or continue straight ahead.
Suppose an experiment consists of observing the movement of one vehicle at
this intersection, and do the following.
• List the elements of a sample space.
• Attach probabilities to these elements if all possible outcomes are
equally likely.
Example 4
# of ways A can occur
( )
# of different simple events
n
P A
N
 

Find the probability that a randomly selected car in U.S. will be in a crash this
year. 6,511,100 cars crashed among the 135,670,000 cars registered. Ans:
0.048
Example 5
When studying the effect of heredity on height, we can express each
individual genotype, AA, Aa, aA, and aa, on an index card and shuffle the four
cards and randomly select one of them. What is the probability that we select
a genotype in which the two components are different? Ans: 0.5
Probability axioms
1. 0  P(A)  1
The probability of an impossible event is 0.
The probability of an event that is certain to occur is 1.
2. P (S ) = 1
Complement (non-Probability)
The Complement Rule states that the sum of the probabilities of an event
and its complement must equal 1.
P(A) + P(A)c
) = 1   c
A A A
Complement of an event is that the event did not occur. = not A. e.g., if A=
red card, Then is a black card (not a red card).
This axiom says that the probability of everything in
the sample space is 1. This says that the sample space
is complete and that there are no sample points or
events that allow outside the sample space that can
occur in our experiment.

Example 7
Consider the experiment of tossing a coin ten times. What is the probability
that we will observe at least one head?
Example 8
The General Motors Corporation wants to conduct a test of a new model of
Corvette. A pool of 50 drivers has been recruited, 20 or whom are men. When
the first person selected from this pool, what is the probability of not getting
a male driver?
Example 9
A typical question on a SAT test requires the test taker to select one of five
possible choices: A, B, C, D, or E. because only one answer is correct, if you
make a random guess, your probability of being correct is 1/5 or 0.2. Find the
probability of making a random guess and not being correct (or being
incorrect)
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.
Finding the Probability of “At Least One”
To find the probability of at least one of something, calculate the probability
of none, then subtract that result from 1. That is,
P (at least) =1-P (non)
Example 10
Find the probability of a couple having at least 1 girl among 3 children. Assume
that boys and girls are equally likely and that the gender of a child is
independent of any other child.
Example 11
If the probabilities that an automobile mechanic will service 3, 4, 5, 6, 7, or 8
or more cars on any given workday are, respectively, 0.12, 0.19, 0.28, 0.24,

0.10, and 0.07, what is the probability that he will service at least 5 cars on his
next day at work?
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).

Example 12
Suppose that there were 120 students in the classroom, and that they could be
classified as follows:
Brown Not Brown
Male 20 40
Female 30 30
A: brown hair
P(A) = 50/120
B: female
P(B) = 60/120
P(AB) = P(A) + P(B) – P(AB)

= 50/120 + 60/120 - 30/120
= 80/120 = 2/3
Part 2
When two events A and B are mutually exclusive, P(AB) = 0
And
P(AB) = P(A) + P(B).
A: male with brown hair
P(A) = 20/120
B: female with brown hair
P(B) = 30/120
A and B are mutually exclusive, so that
P(AB) = P(A) + P(B)
= 20/120 + 30/120
= 50/120
Example 13
1. What is the probability of getting a total of 7 or 11 when pair of fair dice is
tossed?
2. 2 fair dice are rolled. What is the probability of getting a sum less than 7
or a sum equal to 10?
Example 14
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?

Example 15
60%of the students at a certain school wear neither a ring nor a necklace. 20%
wear a ring and 30%wear a necklace. If one of the students is chosen
randomly, what is the probability that this student is wearing
3. (a) A ring or a necklace?
4. (b) A ring and a necklace?
Example 16
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?
For three events A, B, and C,
P (A ∪ B ∪ C) =P (A) + P (B) + P (C) −P (A ∩ B) − P (A ∩ C ) − P (B ∩ C )+P (A ∩ B ∩ C).
Example 17
An instructor of a statistics class tells students that the probabilities of
earning an A, B, C, and D or below are 1/5, 2/5, 3/10, &, and 1/10, respectively.
Find the probabilities of (1) earning an A or B and (2) earning a B or below.
If the probabilities are, respectively, 0.09, 0.15, 0.21, and 0.23 that a person
purchasing a new automobile will choose the color green, white, red, or blue,
what is the probability that a given buyer will purchase a new automobile that
comes in one of those colors.
Solution:
Let G, W, R, and B be the events that a buyer selects, respectively, a green,
white, red, or blue automobile. Since these four events are mutually exclusive,
the probability is
P (G∪W∪R∪B) =P (G) +P (W) +P(R) +P (B)
=0.09 + 0.15 + 0.21 + 0.23 = 0.68.

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 (B|A): pronounced "the probability of B given A.”
Example 18 :
Roll a dice. What is the chance that you would get a 6, given that you’ve gotten
an even number?
Example 19:
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?
Example 20:
In a certain city in the USA some time ago, 30.7% of all employed female
workers were white-collar workers. If 10.3% of all workers employed at the
city government were female, what is the probability that a randomly selected
employed worker would have been a female white-collar worker?
 
 
 
|
P A B
P B A
P A



Example 21:
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
Example 22:
The probability that a regularly scheduled flight departs on time is P(D)=0.83;
the probability that it arrives on time is P(A)=0.82; and the probability that it
departs and arrives on time is P(D∩A)=0.78. Find the probability that a plane ;
a) arrives on time, given that it departed on time, Ans =0.94
b) Departed on time, given that it has arrived on time. Ans=0.95
Example 23:
The king comes from a family of 2 children. What is the probability that
the other child is his sister? ans=2/3
Example 24:
A couple has 2 children. What is the probability that both are girls if the
older of the two is a girl? ans= ½
Example 25
A total of 28 percent of American males smoke cigarettes, 7 percent smoke
cigars, and 5 percent smoke both cigars and cigarettes. What percentage of
males smokes neither cigars nor cigarettes?

Multiplication Rule
The multiplication rule is a result used to determine the probability that two
events, A and B, both occur. The multiplication rule follows from the definition
of conditional probability. The result is often written as follows, using set
notation:
P (A ∩ B) = P (A|B) × P (B) or P (B ∩ A) = P (B|A) × P (A)
Theorem
Two events A and B are independent if and only if
P ( A ∩ B) = P (A) P (B).
Therefore, to obtain the probability that two independent events will both
occur, we simply find the product of their individual probabilities.
Flowchart
Example 26:
If P(C)= 0.65, P(D)= 0.4, and P(C D )=0.26, are the event C and D independent ?
Example 27:
If the probability is 0.25 that the person will name red as his/her favourite
colour, what is probability that three totally unrelated persons will all name
red as their favourite colour?

Example 28:
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.
Example 29:
The great composer Ludwig Van Beethoven wrote 9 symphonies and 32 piano
concertos. If an orchestra conductor randomly selects two pieces of music,
without replacement from collection of those 41 pieces what is probability
that:
a) First piece selected is symphony,, and the second piece selected is a
piano concerto
b) Both piece are symphony …..
c) Both piece piano concerto
Example 30:
A jury consists of 9-persons who are native born and 3-person who are foreign
born. If two of the jurors are randomly picked for an interview, what is the
probability that they will both be foreign born?
Example 31:
The probability that an American industry will Locate in Shanghai, Chinais0.7,
the probability that it will locate in Beijing, Chinais0.4, and the probability that
it Will locate in cither Shanghai or Beijing or both is0.8.What is the
probability that the industry will locate
In both cities?
In neither city?
Example 32:
The probability that a doctor correctly diagnoses a particular illness is 0.7.
Given that the doctor makes an incorrect diagnosis, the probability that the
patient files a lawsuit is 0.9. What is the probability that the doctor makes an
incorrect diagnosis and the patient sues?

Example 33:
The probability that a married man watches a certain television show is 0.4,
and the probability that a married woman watches the show is 0.5. The
probability that a man watches the show, given that his wife does, is 0.7. Find
the probability that
(a)a married couple watches the show;
(b)a wife watches the show, given that her husband does;
(c)at least one member of a married couple will watch the show
Example 34:
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?
Example 35:
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?
that both are girls if the older of the two is a girl?ans=1/2

Homework
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
c) (A∩C) ∪ B.
3. Registrants at a large convention are offered 6 sightseeing tours on each of
3 days. In how many ways can a person arrange to go on a sightseeing tour
planned by this convention? Ans=18 ways for a person to arrange a tour.
4. In how many different ways can a true-false test consisting of 9 questions
be answered? Ans =29
5. A developer of a new subdivision offers a prospective home buyer a choice
of 4 designs, 3 different heating systems, a garage or carport, and a patio
or screened porch. How many different plans are available to this buyer?
Ans =48
6. A contractor wishes to build 9 houses, each different in design. In how
many ways can he place these houses on a street if 6 lots are on one side of
the street and 3 lots are on the opposite side? Ans = 362, 880
7. Four married couples have bought 8 seats in the same row for a concert. In
how many different ways can they be seated
a) With no restrictions? Ans = 40320
b) If each couple is to sit together? = 384 ways.

c) if all the men sit together to the right of all the women? = 576 ways
8. If a multiple-choice test consists of 5 questions, each with 4 possible
answers of which only 1 is correct,
a) in how many different ways can a student check off one answer to each
question?
b) in how many ways can a student check off one answer to each question
and get all the answers wrong?
9. If a letter is chosen at random from the English alphabet, find the
probability that the letter
(a) is a vowel exclusive of y;
(b) is listed somewhere ahead of the letter j;
(c) is listed somewhere after the letter g
10.An experiment involves tossing a pair of dice, one green and one red, and
recording the numbers that come up. If x equals the outcome on the green
die and y the outcome on the red die, describe the sample space S by listing
the elements (x, y);
11. Two jurors are selected from 4 alternates to serve at a murder trial. Using
the notation A1 A3, for example, to denote the simple event that alternates
1 and 3 are selected, list the 6 elements of the sample space S.
12.Four students are selected at random from a chemistry class and classified
as male or female. List the elements of the sample space S1, using the
letter M for male and F for female. Define a second sample spaceS2 where
the elements represent the number of females selected.
13.Construct a Venn diagram to illustrate the possible intersections and unions
for the following events relative to the sample space consisting of all

automobiles made in the United States. F: Four door, S: Sun roof, P: Power
steering.
14.Which of the following pairs of events are mutually exclusive?
a) A golfer scoring the lowest 18-hole round in a 72-hole tournament and
losing the tournament.
b) A poker player getting a flush (all cards in the same suit) and 3 of a kind
on the same 5- card hand.
c) A mother giving birth to a baby girl and a set of twin daughters on the
same day.
d) A chess player losing the last game and winning the match.
15.An urn contains 6 red marbles and 4 black marbles. Two marbles are
drawn without replacement from the urn. What is the probability that both
of the marbles are black?
16.Registrants at a large convention are offered 6 sightseeing tours on each of
3 days. In how many ways can a person arrange to go on a sightseeing tour
planned by this convention? Ans=18 ways for a person to arrange a tour.
17.In how many different ways can a true-false test consisting of 9 questions
be answered?

3.2 probablity

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