Matakuliah
Tahun
: F0892 - Analisis Kuantitatif
: 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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What Is A Probability? (continued)
• The probability of 1 represents something that is certain
to happen.
• The probability of 0 represents something that cannot
happen.
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What Is A Probability? (continued)
• 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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Approach To Assigning Probabilities
• Two approach:
1. Objective probability.
2. Subjective probability.
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Approach To Assigning Probabilities
(continued)
Objective probability is subdivided into:
1.
2.
Classical probability.
Empirical probability.
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Approach To Assigning Probabilities
(continued)
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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Approach To Assigning Probabilities
(continued)
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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•
Some Rules
For Computing Probabilities
Rules of addition:
- 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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Some Rules
For Computing Probabilities
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
Number Of
Packages
Probability
Of
Occurrence
Underweight
A
100
0.025
100/4000
Satisfactory
B
3600
0.9
3600/4000
Overweight
C
300
0.075
300/4000
TOTAL
4000
1.00
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* 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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Venn Diagram Represents The
Mutually Exclusive
Event A
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Event B
Event C
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Complement Rule
P(A) = 1 – P(-A)
Event A
-A
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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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