Probability

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EMIS 7370/5370 STAT 5340
NTU MA-520-N
Probability and Statistics for Scientists and
Engineers
Probability Counting Techniques
UPDATED 9/8/03
Dr. Jerrell T. Stracener, SAE Fellow
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• Counting Techniques
• Product Rule
• Tree Diagram
• Permutations
• Combinations

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• 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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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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Tree Diagrams Selection Plumbing Electrical
Outcome Contractors Contractors
P1 E1P1
E1
P2 E1P2
P1 E2P1
E2
P2 E2P2
P1 E3P1
E3
P2 E3P2
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• 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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Example Five identical size books are
available for return to the book shelf. There
are only three spaces available. In how
many ways can the three spaces be filled?
Example Five different books are available
for weekend reading. There is only enough time
to read three books. How many selections can be
made?