#### 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 Leadership in Engineering 1 Stracener_EMIS 7370/STAT 5340_Fall 08_08.27.08 Probability-Counting Techniques • Product Rule • Tree Diagram • Permutations • Combinations 2 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. 3 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? 4 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 5 Tree Diagrams Selection Electrical Plumbing Contractors Contractors E1 E2 E3 Outcome P1 E1P1 P2 P1 E1P2 E2P1 P2 P1 P2 E2P2 E3P1 E3P2 6 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)! 7 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 )! 8 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 9 Examples 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? 10 Stracener_EMIS 7370/STAT 5340_Fall 08_08.27.08