Section 4.3 Trees and Counting Techniques Organize outcomes in a sample space using tree diagrams. Define and use factorials. Explain how counting techniques relate to probability in everyday life Notes: Monday, Nov. 25

Multiplication Rule of Counting In the last section, the # of outcomes in a sample space were small or simple to calculate. In this section, we will be able to count the number of possible outcomes in larger sample spaces or ones formed by more complicated events.

Multiplication Rule of Counting

Example Problems: Mult. Rule of Counting – More than 2 Events 1.Computer Design. In designing a computer, if a byte is defined to be a sequence of 8 bits and each bit must be a 0 or 1, how many different bytes are possible? 2.Chance and Skill. Bob claims that he has the ability to roll a die in such a way that 6 will almost always occur. You test him by giving him a fair die, which he proceeds to roll five times, getting a 6 each time. If Bob has no control over the die, how many outcomes are possible with five rolls of a die? If Bob does get five 6s in five rolls, does it appear that he has control of the die?

3.Jacqueline is in a nursing program and is required to take a course in psychology and one in physiology (A and P) next semester. She also wants to take Spanish II. If there are two section of psychology, two of A and P, and three of Spanish II, how many different class schedules can Jacqueline choose from? (Assume that the times of the sections do not conflict.) Example Problems: Mult. Rule of Counting & Tree Diagram Tree Diagram For Selecting Class Schedules: Table For Schedules Utilizing Section 1 of Psychology: Psych Section A and P Section Spanish II Section

Example: Tree Diagram & Probability Suppose there are 5 balls in a bin. 3 balls are red, 2 are blue. You are to draw out one ball, note its color and set it aside. Continue drawing balls, noting the color and setting them aside. What are the outcomes of the experiment? What is the probability of each outcome?

Survey Questions. When designing surveys, pollsters sometimes try to minimize a lead-in effect by rearranging the order in which the questions are presented. If Gallup plans to conduct a consumer survey by asking subjects 5 questions, how many different versions of the survey are required if all possible arrangements are included? Example Problems: Mult. Rule of Counting & Factorials Factorial Symbol: ! Denotes the product of decreasing positive whole numbers. By special definition, 0! = 1. Factorial on TI-83 Plus: Enter the number, press MATH and select PRB and menu item 4.

A collection of n different items can be arranged in order n! different ways. (This factorial rule reflects the fact that the first item may be selected n different ways, the second item may be selected n – 1 ways, and so on.) Factorial Rule Routing (shortest possible routes):

Air Routes. You have just started your own airline company called Air America (motto: “Where your probability of a safe flight is greater than zero”). You have one plane for a route connecting Austin, Boise, and Chicago. How many routes are possible? Example Problem: Factorial Rule

Homework A#4.31 Page 160 Numbers 5 – 7 and 9 – 12 Due Tuesday, November 26