Chapter 4: Probability

Chapter 4: ProbabilityWhat is probability?A value between zero and one that describe the

relative possibility(change or likelihood) an event occurs.

The MEF announces that in 2012 the change Cambodia economic growth rate is equal to 7% is 80%.

The weather forecast announces that there is a 70 % change of rain for Phnom Penh Saturday.

The 3 keywords used in probability:Experiment: A process that leads to the

occurrence of one and only one of possible outcomes.

In probability, an experiment has two or more results/outcomes and it is uncertain which will occur.

Outcome: A particular result of an experimentEvent: A subset or collection of one ore more

outcomes of experiment

Example: Rolling a die

Experiment Roll a die

a 1

a 2

a 3

All possible outcomes a 4

a 5

a 6

Get an even number

Get a 3 or less

Some possible events Get A number greater than 5

Types of Probability

Types of Probability

Objective Probability

Classical Probability

Empirical Probability

Subject Probability

Classical Probability: The probability that is based on the assumption that the outcomes of an experiments are equally likely.

Example: Consider the experiment of rolling a six-sided die. What is the probability of the event “an odd number of spots face up”?Solution: Possible outcomes are:

a 1-spot a 2-spot a 3-spota 4-spot a 5-spot a 6-spot

The favorable(odd number of spots) outcomes are: a 1-spot a 3-spot a 5-spotTherefore, the probability of an odd number of spots face up is:Probability=3/6=0.5

Mutually excusive: The occurrence of any one event means that none of other events can occur at the same time. In rolling a die experiment, the event of “an odd number of spots face up” and the event of “an even number of spots face up” can not occur at the same time.

Empirical Probability: The probability that is based on the empirical conception of probability.

It is much influenced by the similar events happened in the past.

Example: A study of 800 information technology graduates at the IIC university revealed that 235 of the 800 were not employed in their major area of study in college. For example, a person who majored in computer network is now the marketing manager of a Luxury company.

What is the probability that a particular information technology graduate will be employed in the area other than his or her college major?

Solution:Probability=P(A)=235/800=0.29

P(A) means the probability of an event A that a particular information technology graduate will be employed in the area other than his/her college major.

Subjective Probability: The probability that is based on evaluating available opinions and other information.

It is not or little influenced by the past experience.

Example:-Estimating the change that you will earn an A in this course.-What is the probability that the Honda company will exceed the value 24000 in the next year?

Some rules of Probability:Rules of AdditionThe events must be mutually exclusive If two events A and B are mutually exclusive, the special

rule of addition is:P(A or B)=P(A)+P(B)

Example: An automatic Shaw machine fills plastic bags with a mixture of beans, broccoli, and other vegetables. Most of the bags contain the correct weight, but because of the slight variation in the size of beans and other vegetables, a package might be slightly underweight or overweight. A check of 4,000 bags filled in the past month revealed:

Weight Event Number of Probability of Bags Occurrences

Underweight A 100 0.025Satisfy B 3,600 0.900Overweight C 300 0.075

What is the probability that a particular bag will be underweight or overweight?

Solution: P(A or C)=P(A)+P(C) =0.025+0.075=0.10

A Ven diagram is a useful tool to depict addition and multiplication rules.

Complement Rule: P(A)+P(~A)=1

Event A

Event B

Event C

General Rule of AdditionThe outcomes of experiment may not be

mutually exclusive.P(A or B)=P(A)+P(B)-P(A and B)

Example: What is the probability of randomly chosen card from a card deck will be either the king or the heart?

Solution:P(A or B)=P(A)+P(B)-P(A and B)=4/52+13/52-(1/52)=0.3077

A B

A and B

Rules of MultiplicationSpecial Rule of MultiplicationEvent A and B are independent—The occurrence of

event A does not alter the probability of event B.P(A and B)=P(A)P(B)

Example: Two dies are rolled. What is the probability that

both display the odd number?Solution:

P(A and B)=P(A)P(B) =3/6*3/6=1/4=0.25

General Rule of ProbabilityThe joint probability of events A and B that both

events will happen is found by multiplying the probability of A will happen and the conditional probability of event B will happen.P(A and B)=P(A)P(B|A)where P(B|A) means the probability B will occur given A ready occurred.

Example: Suppose that in a bag there are 2 notes of 10$ and 3 notes of 20$. Two notes are to be selected, one after the other. What is the probability of selecting one note of 10$ followed by one note of 20$?

Solution:P(A and B)=P(A)P(B|A)

=2/5*3/4=0.3

Tree DiagramsThe tree diagram is a graph that is helpful in

organizing calculations that involve several stages.

Each segment of the tree is one stage of the problem

P(B1|A1)=1/4 2/5*1/4=0.1

P(A1)=2/5

P(B2|A1)=3/42/5*3/4=0.3

P(B1|A2)=2/4 3/5*2/4=0.3

P(A2)=3/5

P(B2|A2)=2/4 3/5*2/4=0.3 =1.0

Bayes’ TheoremIn the 18th century, Reverend Thomas Bayes,

and English Presbyterian minister, raised this question: Does God really exist? Interested in mathematics, he tried to develop a formula to find the probability that God really exists based on evidence that was available to him on earth.

Example: Suppose that 10 percent of people in Cambodia

has believed that the God really exists. We will let A1 refer to the event that the God really exists and A2 refer to the event that the God does not exist. Thus, we know that if we select a person from Cambodia randomly, the probability that the individual chosen believes that the God really exist is P(A1)=0.1 and the probability that the individual chosen believes that the God does not exist is P(A2)=1-0.1=0.9. These probabilities are prior probabilities.

Now, let B denote the event “survey shows that the God really exists”. Assume that historical evidence shows that if a person believes the God really exists, the probability that the survey shows that the God really exists is 0.85(P(B|A1). We also assume that if a person believes that the God does not exist, the probability the survey shows that the God really exists is 0.15(P(B|A2).Now select a person from Cambodia randomly and do the survey. The survey shows that his/her believes that the God really exist. What is the probability that the person actually believes that the God really exist?

This probability can be calculated as below:P(A1|B)=P(A1)P(B|A1)/(P(A1)P(B|A1)+P(A2)P(B|A2))

=0.1*0.85/(01*0.85+0.90*0.15)=0.086

=>P(A1|B) is called posterior probability

Principles of CountingLarge number of outcomesPossible arrangements for two or more groupsMultiplication Formula: if there are m ways of

doing one thing and n ways of doing anther thing, there are m*n ways of doing both.

Total number of arrangements=(m)(n)Example: There are 20 male students and 5 female

students. How many different arrangements of male and female students to form a group of two(one male and anther female)?Number of arrangements=20*5=100

The Permutation FormulaPossible arrangements for only one group of

objectsAny arrangement of r objects selected from a

single group of n possible objects can be expressed by the following formula:

nPr=n!/(n-r)!

Example: The 10 numbers from 0 to 9 are to be used in code groups of four to identify an item of clothing. Code 1023 might identify a blue blouse and size small. Code 2051 might identify a T-short, and size medium and so on. The same number can not be used twice or more in each group.

Solution:

nPr=10!/(10-4)!=10!/6! =10*9*8*7=5040

Note: in Permutation, ab and ba are not the same.

The Combination Formula For permutation, the order of objects in their

group makes the group different from all other groups.

If the order of objects in their group does not make the group different from all other groups, the total number of arrangements is called combination.

Combination is the number of ways to choose r objects from a group of n object without regard to order.

Combination Formula:

nCr=n!/r!(n-r)!Example: How many different ways to choose 5

students from 10 students to form a group of five students?

Solution:nCr=10!/5!(10-5)!

=10*9*8*7*6/5*4*3*2*1=252

Note: in Combination ab and ba are same.

Exercises1. Before a nationwide survey, 30 persons were

selected to test a questionnaire. One question about whether NGOs management law is applicable in Cambodia required yes and no answer.

a. What is the experiment?b. List one possible event.c. Twenty of 30 say yes. What is the probability that

a particular person say yes?d. What is the concept of this probability?(Classical, Empirical or subjective probability)

2. The events A and B are mutually exclusive. Suppose P(A)=0.20 and P(B)=.10. What is the probability that either A or B occur?

3. Suppose the probability you will get a grade A in this IT class is 0.30 and the probability that you will get grade B is 0.35. What is the probability that you will get a grade above C? and What is the probability that you will get a grade below B?

4. Mr. Sok is taking two courses, C++ programming and Software Engineering. The probability Sok will pass C++ programming is 0.65 and the probability of passing Software Engineering is 0.70. The probability Sok will pass both is 0.50. What is the probability that Sok will pass at least one course?

5. Suppose P(A)=0.25 and P(B)=0.50. What is the joint probability of A and B?

6. In tossing 3 coins at the same time, what is the probability that the three coin show tails?

7. In the experiment of select two cards( one after another without inserting in to the deck again) from a cards deck. What is the probability that both are kings.

8. Three defective eggs were accidentally sold (pick one after another) at a small shop near Olympic market along with 50 non-defective eggs.

a. What is the probability that the first two eggs were sold are defective?

b. What is the probability that the first two eggs were non-defective?c. Draw a Ven’ diagram to illustrate these probabilities.

9. The board of directors of Luxury company consists of 10 persons in which 3 persons are female. 3 of the board are selected randomly to form a committee.

a. What is the probability that the committee consists of 3 women?

b. What is the probability that the committee consists of at least 1 man?

10.For the daily lottery game in Phnom Penh, each game participant needs to buy at least one ticket. Every ticket contains three numbers between 0 and 9. A number(1 digit) can not be appeared twice or more in the ticket. The winning tickets are announced in CTN TV station every night at 7:30 pm.a. How many different outcomes(3 digits) are possible?b. If you purchase 1 ticket tonight, what is the change that you will win?c. If you purchase 2 tickets to night, what is the change that one of your tickets will win?