NATIONAL MATH + SCIENCE INITIATIVE Mathematics A2T 7.2 Classwork Probability Using Sample Spaces, Permutations, and Combinations 1. a. Before playing the game, do you think this game is fair? Explain your reasoning. b. With a partner, play this game 20 times and record who wins each round. After completing the 20 rounds of the game, create a sample space to show all of the possible outcomes for the game. c. Based on the sample space, is the game fair to both players? If not, how would you change the rules so that both players have an equal probability of winning the game? 2. dice are rolled at the same time, each face has an equal probability of landing on top. Two players are going to play a game and the person with the higher number on top wins the game. a. Create a sample space showing the possible outcomes. b. Based on the sample space, which die would you select to have a greater chance of winning? Explain your reasoning. c. If a 5 is face up, what is the probability that the person with die A win? Copyright © 2013 National Math + Science Initiative, Dallas, Texas. All rights reserved. Visit us online at www.nms.org. 1 Mathematics—Probability Using Sample Spaces, Permutations, and Combinations For questions 1 and 2, the sample spaces are small enough that all of the outcomes can be listed, and the principle, permutations, and combinations can be used to determine the total number of outcomes and the probability that those outcomes occur. can occur in p ways and another event can occur in q ways, then the number of ways that both events can occur is equal to pq . This principle applies to more than two events. For example, if three events can occur in p, q, and r ways, then the number of ways that all three can occur is pqr . 3. a. be W or K and no letters repeat? b. c. A photographer is taking a picture of a bride and groom together with 6 attendants. in a certain order. The number of arrangements of n objects taken r at a time, n Pr , is determined by n! . the formula n Pr n r 4. a. 2 P b. 10 P4 5. Write 25 24 23 22 as a quotient of factorials and as a permutation. 6. Write n P4 as a quotient of factorials and simplify. Copyright © 2013 National Math + Science Initiative, Dallas, Texas. All rights reserved. Visit us online at www.nms.org. Mathematics—Probability Using Sample Spaces, Permutations, and Combinations A2T - 7.2 Classwork 7. students for the demonstration: one student will bounce the ball, one will hold the motion detector and graphing calculator, and one will record the data. a. Assuming a student may hold only one position, how many different ways can the roles be assigned if there are n students? b. c. bounce the ball? d. What is the theoretical probability that you will serve in one of the roles? Assume you have 4 textbooks in your backpack and have decided to arrange three of them side by side on your desk. a. b. Assuming the 4 books include English, Science, Math, and History, write the sample space for the possible arrangements of 3 books selected from the set of 4. c. Your friend has the same 4 textbooks in his locker and has decided to take 3 of the books home for studying. Write the sample space for the possible sets of 3 textbooks that can be selected. d. e. Why is the number of possible outcomes different for these two scenarios? Copyright © 2013 National Math + Science Initiative, Dallas, Texas. All rights reserved. Visit us online at www.nms.org. 3 Mathematics—Probability Using Sample Spaces, Permutations, and Combinations arrangement is not considered. Often this is referred to as the number of sets of n elements chosen r at a time, and for notation written as n Cr . The formula for a combination is . 9. a. C b. 10 C4 10. A club consists of 3 girls and 5 boys. a. b. c. What is the probability that the committee selected consists of 2 girls and 1 boy? 11. In a study of migration patterns of great white sharks off of the southern tip of Africa, 100 sharks were tagged with a yellow or green tag. The sharks were then monitored using a GPS tracking device. The two-way table shows how the tags were distributed among male and female sharks. Yellow Green Total Male 34 20 54 Female 16 30 46 Total 50 50 100 a. b. with green tags? c. selected at random, there are exactly 2 female sharks with green tags? 4 Copyright © 2013 National Math + Science Initiative, Dallas, Texas. All rights reserved. Visit us online at www.nms.org.