Methods of Enumeration

1. Methods of Enumeration • Counting tools can be very important in probability … particularly if you have a finite sample space with equally likely outcomes • Multiplication Principle (or Product Rule) • An experiment consisting of m steps with ni possible outcomes on the ith step has a total of n1× n2× … × nm possible outcomes • Tree Diagram

2. Sampling Terminology • Sampling with Replacement • Sampling where an object is selected and replaced before the next object is selected. The probability of selection remains constant. • Sampling without Replacement • Sampling where an object is not replaced after it is selected. The probability of selection changes after each selection.

3. Order is Important • Ordered Sample of Size r • r objects are selected from n objects, and the order is important • Sampling with Replacement • There are nr possible ordered samples when sampling r objects from a set of n objects with replacement.

4. Order is Important • Sampling without Replacement • There are n! different ways in which n objects can be ordered. Each possible ordering is called a permutation. • Permutation Rule • For r≤n, the number of permutations of r objects selected from n objects is

5. Order is not Important • Order of selection is often not important. If so, then all r! orders of r objects selected from n objects are considered the same. • Sampling without Replacement • Combination Rule • For r≤n, the number of combinations of r objects selected from n objects is

6. Binomial Coefficient • nCr is equivalent to the binomial coefficient, which is the coefficient in the kth expansion of (a+b)n as expressed below: • Pascal’s Triangle

7. Distinguishable Permutations • Binomial Coefficient • Useful for counting the number of distinguishable permutations in a set of two types with r of one type and (n-r) of the other. • Multinomial Coefficient • Userful for counting the number of distinguishable permutations in a set of s types where there are ni objects of the ith type (n=n1+n2+…+ns)