#### Transcript Probability - Counting Techniques

```Systems Engineering Program
Department of Engineering Management, Information and Systems
EMIS 7370/5370 STAT 5340 :
PROBABILITY AND STATISTICS FOR SCIENTISTS AND ENGINEERS
ProbabilityCounting Techniques
Dr. Jerrell T. Stracener, SAE Fellow
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Stracener_EMIS 7370/STAT 5340_Fall 08_08.27.08
Probability-Counting Techniques
• Product Rule
• Tree Diagram
• Permutations
• Combinations
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Stracener_EMIS 7370/STAT 5340_Fall 08_08.27.08
Product Rule
• Rule
If an operation can be performed in n1 ways, and if for each of
these a second operation can be performed in n2 ways, then the
two operations can be performed in n1n2 ways.
• Rule
If an operation can be performed in n1 ways, and if for each of
these a second operation can be performed in n2 ways, and for
each of the first two a third operation can be performed in n3
ways, and so forth, then the sequence of k operation can be
performed in n1, n2, …, nk ways.
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Stracener_EMIS 7370/STAT 5340_Fall 08_08.27.08
Tree Diagrams
Definition:
A configuration called a tree diagram can be used to represent
pictorially all the possibilities calculated by the product rule.
Example:
A general contractor wants to select an electrical contractor and a
plumbing contractor from 3 electrical contractors, and 2 plumbing
contractors. In how many ways can the general contractor choose
the contractor?
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Stracener_EMIS 7370/STAT 5340_Fall 08_08.27.08
Tree Diagrams
Selection
Plumbing Electrical
Contractors Contractors
P1
P2
Outcome
E1
E2
P1E1
P1E2
E3
P1E3
E1
E2
P2E1
P2E2
E3
P2E3
By observation there are 6 ways for the contractor to choose
the two subcontractors. Using the product rule, the number is
2x3=6
Stracener_EMIS 7370/STAT 5340_Fall 08_08.27.08
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Tree Diagrams
Selection
Electrical Plumbing
Contractors Contractors
E1
E2
E3
Outcome
P1
E1P1
P2
P1
E1P2
E2P1
P2
P1
P2
E2P2
E3P1
E3P2
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Stracener_EMIS 7370/STAT 5340_Fall 08_08.27.08
Factorial
• Definition:
For any positive integer m, m factorial, denoted by
m!, is defined to be the product of the first m positive integers, i.e.,
m! = m(m - 1)(m - 2) ... 3  2  1
Rules:
0! = 1
m! = m(m - 1)!
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Stracener_EMIS 7370/STAT 5340_Fall 08_08.27.08
Permutations
• Definition
A permutation is any ordered sequence of k objects taken
from a set of n distinct objects
• Rules
The number of permutations of size k that can be constructed
from n distinct objects is:
n
Pk  n(n  1)(n  2)...( n  k  2)(n  k  1)
n!
n Pk 
(n  k )!
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Stracener_EMIS 7370/STAT 5340_Fall 08_08.27.08
Combinations
• Definition
A combination is any unordered subset of size k taken
from a set of n distinct elements.
• Rules
The number of combinations of size k that can be formed
from n distinct objects is:


 n
n!
  n Ck 
k!(n  k )!
k 
 n  n Pk
  
 k  k!
Stracener_EMIS 7370/STAT 5340_Fall 08_08.27.08
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Examples
Example1
book shelf. There are only three spaces available. In how many
ways can the three spaces be filled?
Example2
Five different books are available for weekend reading.
There is only enough time to read three books. How many