1 Dr. Jerrell T. Stracener, SAE Fellow Leadership in Engineering EMIS 7370/5370 STAT 5340 : PROBABILITY AND STATISTICS FOR SCIENTISTS AND ENGINEERS Systems Engineering Program Department of Engineering Management, Information and Systems Stracener_EMIS 7370/STAT 5340_Fall 08_ Probability- Counting Techniques

Stracener_EMIS 7370/STAT 5340_Fall 08_ Product Rule Tree Diagram Permutations Combinations Probability-Counting Techniques

Stracener_EMIS 7370/STAT 5340_Fall 08_ Rule If an operation can be performed in n 1 ways, and if for each of these a second operation can be performed in n 2 ways, then the two operations can be performed in n 1 n 2 ways. Rule If an operation can be performed in n 1 ways, and if for each of these a second operation can be performed in n 2 ways, and for each of the first two a third operation can be performed in n 3 ways, and so forth, then the sequence of k operation can be performed in n 1, n 2, …, n k ways. Product Rule

Stracener_EMIS 7370/STAT 5340_Fall 08_ 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? Tree Diagrams

Stracener_EMIS 7370/STAT 5340_Fall 08_ Selection Plumbing Electrical Contractors Contractors Outcome E1P1E1E1P1E1 E3P1E3E3P1E3 E2P1E2E2P1E2 E1P2E1E1P2E1 E3P2E3E3P2E3 E2P2E2E2P2E2 P1P1 P2P2 By observation there are 6 ways for the contractor to choose the two subcontractors. Using the product rule, the number is 2 x 3 = 6 Tree Diagrams

Stracener_EMIS 7370/STAT 5340_Fall 08_ Selection Electrical Plumbing Contractors Contractors Outcome P1E1P1P1E1P1 P2E1P2P2E1P2 P1E3P1P1E3P1 P2E3P2P2E3P2 E1E1 E3E3 P1E2P1P1E2P1 P2E2P2P2E2P2 E2E2 Tree Diagrams

Stracener_EMIS 7370/STAT 5340_Fall 08_ 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)! Factorial

Stracener_EMIS 7370/STAT 5340_Fall 08_ 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: Permutations

Stracener_EMIS 7370/STAT 5340_Fall 08_ 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:   Combinations

Stracener_EMIS 7370/STAT 5340_Fall 08_ Example1 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? Example2 Five different books are available for weekend reading. There is only enough time to read three books. How many selections can be made? Examples