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TELENET 2120:
NETWORK PERFORMANCE
Basic Probability
Asst. Prof. Joseph Kabara, Ph.D
Graduate Program in Telecommunications and Networking
University of Pittsburgh
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Spring 2008, Class #2
TELENET 2120: Network Performance
Terminology
•
Probability Theory 
based on the concept of a random
experiment
•
Random
– phenomenon/experiment where an individual
outcome is uncertain but there is a regular distribution of
outcomes in a large number of repetitions.
•
Probability 
proportion of times a specific
outcome would
occur in a long series of repetitions of the experiment.
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Spring 2008, Class #2
TELENET 2120: Network Performance
•
LongTerm Relative Frequency
– If toss a single coin, the relative frequency of heads is erratic for 2, or 5,
or 10 tosses.
– If you toss the coin several thousand times, the relative frequency
remains stable.
•
Mathematical probability is an idealization of what would happen to
the relative frequency after infinite number of repetitions of random
experiment.
n
repetition
n
in
occurs
A
event
times
of
number
frequency
relative
=
What is Probability?
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Spring 2008, Class #2
TELENET 2120: Network Performance
Probability of Heads
Probability based on long
term
term relative frequency!
0.501
501
1000
0.55
11
20
0.60
3
5
0.33
1
3
1.00
1
1
Relative Frequency
# Heads
# Tosses
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Spring 2008, Class #2
TELENET 2120: Network Performance
Probability Model
•
Sample Space
 set of all possible outcomes of a random
experiment (
S
).
•
Event
 an outcome or a set of outcomes of the experiment
•
Probability measure
is a number or function that maps from the
events in the sample space to a real number between 0 and 1
•
The probability of all possible outcomes (that is the sample
space) must equal 1
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Spring 2008, Class #2
TELENET 2120: Network Performance
Probability Model
•
Example:
Toss of a single die
•
Sample Space
:
S =
{1,2,3,4,5,6}
•
Event : A =
{rolled even number},
B =
{rolled odd number},
D=
{rolled a 2}
•
Probability measure:
P(A) = .5, P(B) = .5, P(D)
= 1/6
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Spring 2008, Class #2
TELENET 2120: Network Performance
Probability Rules
•
Remember the probability of any event P(A) must satisfy
0 <
P(A) <
1
•
Complement Rule
The complement of any event A is the chance that A does not occur
P(A
c
) = 1  P(A)
Example: Toss a single die:
S
= {1,2,3,4,5,6};
let A = {2,4},
A
c
= {1,3,5,6};
P(A)
= 1/3;
P(A
c
) = 11/3 = 2/3
•
Addition Rule
For
two events
A and B that are disjoint (no common outcomes)
P (A or B) = P(A) + P (B)
Example: Toss a single die:
S
= {1,2,3,4,5,6};
let A = {2},
B = {1,3,5};
P(A or B) = P(A)
+ P(B) = 1/6 + 1/2 = 2/3
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Spring 2008, Class #2
TELENET 2120: Network Performance
Probability Rules
•
Multiplication Rule
= two events
A and B are independent, if knowing
that one occurs does not change the probability that other occurs
P (A and B) = P(A)*P(B)
Example: Toss a pair of die .
S
= {(1,1),(1,2),….(6,6)} 36 possible outcomes.
Let A ={first die shows 6} = {(6,1),(6,2),(6,3),(6,4),(6,5),(6,6)}
Let B = {second die shows 1} = {(1,1),(2,1),(3,1),(4,1),(5,1),(6,1)}
Then P(A) = 6/36 = 1/6; P(B) = 6/36 = 1/6 and
P(first die 6, second die 1) = P(A and B) = 1/36 = P(A) P(B)
implies independence
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Spring 2008, Class #2
TELENET 2120: Network Performance
Probability Rules
•
Multiplication Rule
Example of Dependent Case: Toss a pair of die
S
= {(1,1),(1,2),….(6,6)} ; 36 possible outcomes
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This document was uploaded on 11/02/2009.
 Spring '09

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