A value between zero and one that describe the relative possibilitychange or likelihood an event occurs The MEF announces that in 2012 the change Cambodia economic growth rate is equal to 7 is 80 ID: 760317
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Slide1
Chapter 4: Probability
What 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.
Slide2The 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 experiment
Event:
A subset or collection of one ore more outcomes of experiment
Slide3Example: Rolling a die
ExperimentRoll a diea 1a 2a 3All possible outcomesa 4a 5a 6Get an even numberGet a 3 or lessSome possible eventsGet A number greater than 5
Slide4Types of Probability
Slide5Classical 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-spot a 4-spot a 5-spot a 6-spot
Slide6The favorable(odd number of spots) outcomes are: a 1-spot a 3-spot a 5-spot
Therefore, 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.
Slide7Empirical 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.
Slide8What 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.
Slide9Subjective 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?
Slide10Some rules of Probability:
Rules of Addition
The 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:
Slide11Weight Event Number of Probability of Bags OccurrencesUnderweight 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
Slide12A 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
Slide13General Rule of Addition
The 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
Slide14A B A and B
Slide15Rules of Multiplication
Special Rule of Multiplication
Event 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
Slide16General Rule of Probability
The 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$?
Slide17Solution:
P(A and B)=P(A)P(B|A)
=2/5*3/4=0.3
Slide18Tree Diagrams
The 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
Slide19P(B1|A1)=1/4 2/5*1/4=0.1 P(A1)=2/5 P(B2|A1)=3/4 2/5*3/4=0.3 P(B1|A2)=2/4 3/5*2/4=0.3P(A2)=3/5 P(B2|A2)=2/4 3/5*2/4=0.3 =1.0
Slide20Bayes’ 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.
Slide21Example:
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.
Slide22Now, 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?
Slide23This 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
Slide24Principles of Counting
Large number of outcomes
Possible arrangements for two or more groups
Multiplication 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
Slide25The Permutation Formula
Possible arrangements for only one group of objects
Any arrangement of
r
objects selected from a single group of
n
possible objects can be expressed by the following formula:
n
P
r
=n!/(n-r)!
Slide26Example: 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:
n
P
r
=10!/(10-4)!=10!/6!
=10*9*8*7=5040
Note: in Permutation,
ab
and
ba
are not the same.
Slide27The 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.
Slide28Combination Formula:
n
C
r
=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.
Slide29Exercises
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.
What is the experiment?
List one possible event.
Twenty of 30 say yes. What is the probability that a particular person say yes?
What is the concept of this probability?
(Classical, Empirical or subjective probability)
Slide30The 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?
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?
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?
Slide31Suppose P(A)=0.25 and P(B)=0.50. What is the joint probability of A and B?
In tossing 3 coins at the same time, what is the probability that the three coin show tails?
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.
Three defective eggs were accidentally sold (pick one after another) at a small shop near Olympic market along with 50 non-defective eggs.
What is the probability that the first two eggs were sold are defective?
What is the probability that the first two eggs were non-defective?
Draw a
Ven
’ diagram to illustrate these probabilities.
Slide32The 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.
What is the probability that the committee consists of 3 women?
What is the probability that the committee consists of at least 1 man?
Slide33For 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?