Matakuliah : F0892 - Analisis Kuantitatif
Tahun : 2009
Probability Theory
Pertemuan 4

Chapter 5. A Survey Of Probability
Concepts
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What Is A Probability?
• The weather forecaster announces that there is a 70 percent chance of rain for Super Bowl Sunday.
• Probability:
A value between zero and one, inclusive, describing the relative possibility (chance or likelihood) an event will occur.
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• The probability of 1 represents something that is certain to happen.
• The probability of 0 represents something that cannot happen.
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• Three key words to study probability:
1. Experiment.
A process that leads to occurrence of one and only one of several possible observations.
2. Outcome.
A particular result of an experiment.
3. Event.
A collection of one or more outcomes of an experiment.
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Example
Experiment: Roll a die
All possible outcome: 1,
2,
3,
4,
5,
6,
Some possible outcome:
- Observe an even number.
- Observe a number greater than 4.
- Observe a number 3 or less.
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• Two approach:
1. Objective probability.
2. Subjective probability.
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Objective probability is subdivided into:
1.
Classical probability.
2.
Empirical probability.
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Classical probability:
- assumption: the outcomes of an experiment are equally likely.
- Equation 5-1 Page 142.
Classical Probability
Probability of an event = (Number of favorable outcomes)
/ (Total number of possible outcome)
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Example
Consider an experiment of rolling a six sided die. What is the probability of the event “an even number of spots appear face up”?
The possible outcome
- A one spot
- A two spot
- A three spot
- A four spot
- A five spot
- A six spot
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Example
There are three favorable outcomes:
- A two
- A four
- A six
Probability of an even number:
3 / 6 = 0.5
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Empirical probability:
- The probability of an event happening is the fraction of the time similar events happened in the past.
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Example
On February 1, 2003, the space shuttle Columbia exploded. This was the second disaster in 113 space missions for NASA. On the basis of this information, what is the probability that a future mission is successfully completed?
A = a future mission is successfully completed
Probability of a successful flight:
(number of successful flight) / (total number of flight)
P(A) = 111 / 113 = 0.98
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- the events must be mutually exclusive
(when one event occurs, none of the other events can occur at the same time).
- Equation 5-2 Page 147.
P(A or B) = P(A) + P(B)
P(A or B or C) = P(A) + P(B) + P(C)
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Example Page 147 and 148.
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 variation in the size of the beans and other vegetables, a package might be underweight or overweight. A check of 4000 packages filled in the past month revealed.
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Weight Event
Underweight A
Satisfactory B
Overweight C
TOTAL
Statistic Descriptive
Number Of
Packages
100
Probability
Of
Occurrence
0.025
100/4000
3600
300
4000
0.9
0.075
1.00
3600/4000
300/4000
17

* What is the probability that a particular package will be either underweight or overweight?
P (A or C) = P(A) + P(C)
= 0.025 + 0.075 = 0.10
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Statistic Descriptive
Event A
Event B
Event C
19

Statistic Descriptive
Event A
-A
20

Some Rules
For Computing Probabilities (continued)
• Joint probability (not mutually exclusive):
- A probability that measures the likelihood two or more events will happen concurrently.
P(A or B) = P(A) + P(B) – P(A and B)
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Example
What is the probability that a card chosen at from a standard deck of cards will be either a king or heart?
A deck of card is 52 cards
A king, there are 4 cards (A)
A heart, there are 13 cards (B)
King…………………...P(A): 4/52
Heart……….…………P(B): 13/52
King of Heart…………P(A and B): 1/52
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Example (continued)
P(A or B) = P(A) + P(B) – P(A and B)
= 4/52 + 13/52 - 1/52
= 16/52
Figure Page 151
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Some Rules
For Computing Probabilities (continued)
• Special rule of multiplication (independence):
- Two events A and B are independent.
- Two events are independent if the occurrence of one event does not alter the probability of the occurrence of the other event.
P(A and B) = P(A) P(B)
P(A and B and C)= P(A) P(B) P(C)
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Example
A survey by the American Automobile
Association (AAA) revealed 60 percent of its members made airline reservations last year.
Two members are selected at random. What is the probability both made airline reservations last year?
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Example (continued)
R1 = the first member made an airline reservation last year
R2 = the second member made an airline reservation last year
Assuming R1 and R2 are independent
P(R1) = 0.6
P(R2) = 0.6
P(R1 and R2) = P(R1) P(R2) = (0.6) (0.6) = 0.36
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Some Rules
For Computing Probabilities (continued)
• General rule of multiplication (not independent):
- Two events are not independent, they are referred to as dependent.
- Conditional probability, the probability of a particular event occurring, given that another event has occurred.
P(A and B) = P(A) P(B I A)
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Example
A golfer has 12 golf shirts in his closed.
Suppose 9 of these shirts are white and the others are blue. He gets dressed in the dark, so he just grabs a shirt and puts it on. He plays golf two days in a row and does not do laundry.
What is the likelihood both shirts selected are white?
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Example (continued)
W1 = the event that the first selected is white
P(W1) = 9/12
W2 = the event that the second selected is white
P(W2 I W1) = the conditional probability
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Contingency Tables
• Contingency tables
A table used to classify sample observations according to two or more identifiable characteristics.
Example Page 156.
Example P
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Tree Diagrams
• A graph that is helpful in organizing calculations that involve several stages.
• Each segment in the tree is one stage of problem.
• The branches of a tree diagram are weighted by probabilities.
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Tree Diagrams (continued)
• Example:
Table 5-1 Page 157 (Lind)
Chart 5-2 Page 159 (Lind)
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Principles Of Counting
• The Multiplication Formula: if there are “m” ways doing one thing and “n” ways of doing another thing, there are “m x n” ways of doing both.
Equation 5-8 Page 165 (Lind)
Example:
Page 165 and 166 (Lind)
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Principles Of Counting (continued)
• The Permutation Formula: any arrangement of r objects selected from a single group of n possible objects.
The order is very important.
Equation 5-9 Page 167 (Lind).
Example:
Page 167 (Lind)
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Principles Of Counting (continued)
• The Combination Formula: the number of r object combinations from a set of n objects. Order of the selected is not important.
Equation 5-10 Page 168 (Lind).
Example:
Page 168 (Lind)
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