Predicting Outcomes of Events Skill 20

Objectives Organize outcomes in a sample space using tree diagrams. Compute number of ordered arrangements of outcomes using permutations. Compute number of (nonordered) groupings of outcomes using combinations. Explain how counting techniques relate to probability in everyday life.

Trees and Counting Techniques When outcomes are equally likely, we compute the probability of an event by The probability formula requires that we be able to determine the number of outcomes in the sample space.

Trees and Counting Techniques When an outcome of an experiment is composed of a series of events, the multiplication rule gives us the total number of outcomes.

Example–Multiplication Rule Suppose you are required to take a course in psychology and one in physics next semester. You also want to take Calculus. If there are two sections of psychology, two of physics, and three of Calculus, how many different class schedules are there to choose from? (Assume that the times of the sections do not conflict.)

Example–Solution Creating a class schedule can be considered an experiment with a series of three events. Two possible outcomes for psychology, Two for the physics, and Three for the Calculus. By the multiplication rule, the total number of class schedules possible is 2  2  3 = 12

Trees and Counting Techniques A tree diagram gives a visual display of the total number of outcomes of an experiment consisting of a series of events. From a tree diagram, we can determine not only the total number and the individual outcomes.

Example–Tree Diagram Using the information from Example 9, make a tree diagram that shows all the possible course schedules. Solution: Tree Diagram for Selecting Class Schedules

Trees and Counting Techniques

Trees and Counting Techniques To find the number of ordered arrangements of n objects taken as an entire group. Use the permutation formula.

Example-Permutations Rule Compute the number of possible seating arrangements for 9 people in 4 chairs. Solution: In this case, consider a total of n = 9 different people, and arrange r = 4 of these people. 𝑃 9,4 = 9! 9−4 ! = 9! 5! = 9∗8∗7∗6∗5! 5! =3024

Example–Solution Using the multiplication rule, we get the same results Permutations rule has the advantage of using factorials.

Trees and Counting Techniques In each of our previous counting formulas, take the order of the objects or people into account. When the order is not important the goal is to find all possible different groupings or combinations of a set information.

Trees and Counting Techniques

Example–Combinations In your math class, you are assigned to watch any 3 math history movies from a list of 10 films. How many different groups of 3 are available from the list of 10? Solution: In this case, use combinations, rather than permutations, of 10 books taken 4 at a time. Using n = 10 and r = 3, we have 𝐶 10,3 = 10! 3! 10−3 ! = 10! 3!7! = 10∗9∗8∗7! 3∗2∗1∗7! There are 120 different groups of 3 films that can be selected from the list of 10.

Trees and Counting Techniques

20: Predicting Outcomes of Events Summarize Notes Questions? Homework Worksheet