1 Probability Counting Techniques Dr. Jerrell T. Stracener, SAE Fellow EMIS 7370/5370 STAT 5340 NTU MA-520-N Probability and Statistics for Scientists and Engineers UPDATED 9/8/03

2 Counting Techniques Product Rule Tree Diagram Permutations Combinations

3 Product Rule 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.

4 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?

5 Tree Diagrams: Selection PlumbingElectrical OutcomeContractors 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

6 Tree Diagrams: Selection PlumbingElectrical OutcomeContractors P1E1P1P1E1P1 P2E1P2P2E1P2 P1E3P1P1E3P1 P2E3P2P2E3P2 E1E1 E3E3 P1E2P1P1E2P1 P2E2P2P2E2P2 E2E2

7 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)!

8 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:

9 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:  

10 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?