Combinatorics Chapter 3 Section 3.1 Trees and Equally Likely Outcomes Finite Mathematics – Stall

3.1 Trees and Equally Likely Outcomes Combinatorics: _____The study of counting___ List two examples of counting problems given on page ____________________________________ 2._____________________________________ Equally Likely Outcomes

3.1 Trees and Equally Likely Outcomes Give an example of a one stage experiment. ________________________________________________ All the possible outcomes of an experiment is called the __________________________________ of the experiment. A simple two stage experiment might be rolling a die and then flipping a coin and noting the results. The sample space for this experiment contains ordered pairs of elements. List them: S = {1H, 2H, ___________________________} This is the Sample Space for this two stage experiment.

Example A: A committee of four people must select a chairman and a secretary for the next year. How many outcomes are in the sample space? Solution: Let the four people be represented by the letters a, b, c, and d. The first letter can be the person selected to be chairman, and the second letter is the person selected as secretary. The sample space S = __________________________________ Note: Elements of the sample space “ab” is not the same as “ba”!!! This represents the same two people but in ____________ Equally Likely Outcomes

Work problems 1-4 from page _________________________________________________________ __________________________________________________________ 2. ___________________________________________________________ ___________________________________________________________ 3. _______________________________________________________________ _______________________________________________________________ 4. _________________________________________________ __________________________________________________________ Equally Likely Outcomes

Tree Diagrams: When are tree diagrams most useful? ____________________________________ ___________________________ Equally Likely Outcomes

Example B: An experiment consists of selecting one card from a deck and noting the suit, then flipping a coin and noting heads or tails. Draw a tree diagram to represent the outcomes. Solution: Work problems 5 – 9 on page 66. Equally Likely Outcomes

Apply the Fundamental Counting Principle to Example B Stage one could be Hearts, Diamonds, Spades or Clubs and Stage two could be _________________ So the number of possible outcomes is 4 ∙ ____ = _____ Equally Likely Outcomes

Example C: A history test consists of 4 T/F questions followed by 3 multiple-choice questions which contain 5 responses to each. Each question on the test has only one correct response. How many different ways can a student respond to the seven questions? Assume no questions are left blank. Solution: Would drawing a tree diagram be practical? Why or why not?____________________________________ Solution: Try the Fundamental Counting Principle: 2x2x2x2x5x5x5 = 2,000 different ways a student can answer the seven questions. Only one of the 2,000 ways makes a perfect score!! Equally Likely Outcomes

Work problems on page 67. Equally Likely Outcomes

Example D: A hat contains the names of four people ( Amy, Bart, Cody, and Duke) who are entered in a contest. A name will be drawn from the hat and that person will win $100. Then a second name will be drawn from the same hat and this person will receive $5. How many ways can the two cash prizes be awarded to the four people entered in the contest? Solution: What is the number of outcomes for the first stage? ___________ What is the number of outcomes on each branch of the first stage representing the second stage? ______________________ Draw the tree. There are 12 different outcomes. Using the Fundamental Counting Principle, we would write 4 ∙ 3 = 12 Equally Likely Outcomes

***Change the above problem to awarding four prizes to all four people. Using the FCP, we write for 4 stages. 4 ∙ 3 ∙ 2 ∙ 1 = 24 This computation can be written as 4!, read as “four Factorial”. Equally Likely Outcomes

Example E Equally Likely Outcomes

Example F Equally Likely Outcomes

3.1 IP page 71 – 72: problems #17-27 odd, 28 – 31. Questions?? 3.1 B IP page 71 – 72: problems #16-30 even. Equally Likely Outcomes