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

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

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

nP 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 cA 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(AB) = P(A) + P(B) – P(AB)




= 50/120 + 60/120 - 30/120

= 80/120 = 2/3

Part 2

When two events A and B are mutually exclusive, P(AB) = 0

And

P(AB) = 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(AB) = 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 BP 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?