MATH S121 A Foundation In Pure Mathematics (Free Courseware) © The Open University of Hong Kong This work is licensed under a Creative Commons-ShareAlike 4.0 International License Contents Chapter 1 Counting and Basic Probability ..................................................................1 1.1 About this module..............................................................................................................1 1.2 Introduction ........................................................................................................................2 1.3 Counting .............................................................................................................................2 1.3.1 Number of outcomes of an event .........................................................................2 1.3.2 Addition Principle ....................................................................................................3 1.3.3 Multiplication Principle ...........................................................................................4 1.3.4 Permutations and combinations...........................................................................6 1.3.4.1 Permutation..................................................................................................7 1.3.4.2 Activity 1 .......................................................................................................8 1.3.4.2.1 Activity 1 feedback ............................................................................8 1.3.4.3 Combinations ...............................................................................................8 1.3.4.4 Activity 2 .....................................................................................................10 1.3.4.4.1 Activity 2 feedback ..........................................................................10 1.4 Probability .........................................................................................................................11 1.4.1 The probability of one event................................................................................15 1.4.1.1 Internet activity...........................................................................................19 1.4.1.2 Exercise 1 ...................................................................................................20 1.4.1.2.1 Feedback - Exercise 1 .....................................................................20 1.4.1.3 Exercise 2 ...................................................................................................20 1.4.1.3.1 Feedback - Exercise 2 .....................................................................21 1.4.1.4 Exercise 3 ...................................................................................................21 1.4.1.4.1 Feedback - Exercise 3 .....................................................................21 1.5 Conclusion.........................................................................................................................21 1 Chapter 1 Counting and Basic Probability 1.1 About this module Available under Creative Commons-ShareAlike 4.0 International License (http:// creativecommons.org/licenses/by-sa/4.0/). Welcome to this free courseware module ‘Counting and Basic Probability’! This module is taken from the OUHK course MATH S121 A Foundation in Pure Mathematics (http://www.ouhk.edu.hk/wcsprd/Satellite?pagename=OUHK/ tcGenericPage2010&lang=eng&ncode=MATH%20S121&shopping=Y&TYPE=CI& CODE=M121),a ten-credit, Foundation level course that is compulsory for a large number of Science and Technology programmes and an option in a number of other programmes, offered by the School of Science and Technology (http://www.ouhk.edu. hk/wcsprd/Satellite?pagename=OUHK/tcSubWeb&l=C_ST&lid=191133000200& lang=eng) of the OUHK. MATH S121 aims to help students develop their skills in handling mathematical activities as well as their understanding of the rigour of mathematics for pursuing middle and higher-level courses in mathematics, computing, and engineering. MATH S121 is mainly presented in printed format and comprises 13 study units. Each unit contains study content, activities, examples, exercises, self-tests, assigned readings, etc for students’ self-learning. This module (The materials for this module, taken from the print-based course MATH S121, have been specially adapted to make them more suitable for studying online, and multimedia elements have been added where appropriate. In addition to this topic on ‘Counting’ and ‘The probability of one event’, which is an extract from Block III Unit 2 of the course, the original Unit 2 also includes the topic ‘The probability of events’, and ‘Conditional probability’.) retains most of these elements, so you can have a taste of what an OUHK course is like. Please note that no credits can be earned on completion of this module. If you would like to pursue it further, you are welcome to enrol in MATH S121 A Foundation in Pure Mathematics (http://www.ouhk.edu.hk/wcsprd/Satellite?pagename=OUHK/ tcGenericPage2010&lang=eng&ncode=MATH%20S121&shopping=Y&TYPE=CI& CODE=M121). This module will take you about twelve hours to complete, including the time for completing the activities and exercises. Good luck, and enjoy your study! 2 1.2 Introduction Available under Creative Commons-ShareAlike 4.0 International License (http:// creativecommons.org/licenses/by-sa/4.0/). Everyone in Hong Kong has heard of the Mark Six! Players select six different numbers from 49 possibilities in the hope of winning a small fortune - or possibly even a large one. So how many different choices of six numbers out of 49 are there? If you just sat down to count out all the possible combination, it would take you a very long time indeed. But in mathematics we have other counting tools that can assist us in this counting task. Learning to use these tools is the basis for learning about counting and probability. The first section in this module introduces basic counting principles, and helps you work through some simple counting problems. You will learn how the Addition and Multiplication Principles can be applied to determine the number of possible combinations. You will then look at two important counting tools, namely permutations and combinations, and compare the different counting principle between them. The second section in this module introduces you to probability as a measure of chance. It gives you a numerical means of comparing different degrees of chance. You learn by looking at common games of chance: tossing a coin, selecting a playing card, or throwing dice. 1.3 Counting Available under Creative Commons-ShareAlike 4.0 International License (http:// creativecommons.org/licenses/by-sa/4.0/). Before you can study the properties of probability, you need to understand the basic counting principles which is a fundamental mathematical idea and an essential part of probability. In this section, you'll develop the concept of two counting principles and use them to determine the number of different outcomes of a certain event, without having to list all of the elements. 1.3.1 Number of outcomes of an event Available under Creative Commons-ShareAlike 4.0 International License (http:// creativecommons.org/licenses/by-sa/4.0/). In general, we use a letter "E" to represent "an event", and n(E) to represent "the number of outcomes of an event E". Let's look at a simple example. We may have an event E defined as 3 Since there are 7 days in a week, in this example the number of outcomes of an event E" is 1.3.2 Addition Principle Available under Creative Commons-ShareAlike 4.0 International License (http:// creativecommons.org/licenses/by-sa/4.0/). Let's look at a simple counting problem. Example 1 Given a small set of numbers denoted by A = {-4, -2, 1, 3, 5, 6, 7, 8, 9, 10}, it is easy to see that there are a total of 10 numbers in A. Now, if we are interested in determining the number of elements of the events E1, E2, and E, which are defined as: • E1 = choosing a negative number from A • E2 = choosing an odd number from A • E = choosing a negative or an odd number from A. Apply the Addition Principle to determine the number of outcomes of events E using E1 and E2. Solution E1 = {-4, -2}, and the number of outcomes for E1 is n(E1) = 2. E2 = {1, 3, 5, 7, 9}, and the number of outcomes for E2 is n(E2) = 5. Events E1 and E2 are mutually exclusive because there is no common outcomes in the list of E1 and E2. So the number of outcomes of event E is This answer can be confirmed if we can list out all the elements of E. In this example the outcomes of E is This gives n(E) = 7, which is the same answer as above. Addition Principle For a series of mutually exclusive events E1, E2, E3,. ,En, and each event, say Ei has n(Ej) outcomes regardless of the process made on the previous events. Then, the total number of possible outcomes is given by 4 Example 2 In how many ways can a number be chosen from the set such that a. it is a multiple of 3 or 8? b. it is a multiple of 2 or 3? Solution a. Let E1 = multiples of 3: E1 = {3, 6, 9, 12, 15, 18, 21}, so n(E1) = 7. Let E2 = multiples of 8: E2 = {8, 16}, so n(E2) = 2. Events E1 and E2 are mutually exclusive, so n(E) = n(E1) + n(E2) = 7+2 = 9 b. Let E1 = multiples of 2: E1 = {2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22}, so n(E1) = 11. Let E2 = multiples of 3: E2 = {3, 6, 9, 12, 15, 18, 21}, so n(E2) = 7. The events E1 and E2 are not mutually exclusive since they contain the same elements {6, 12, 18}. If the problem