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Statistics · 200 Level
This is the complete lecture note for STA 211 Probability II, prepared for Statistics students at 200 Level. You can read it online below, open it in the full viewer, or download the PDF for offline study. It pairs with the course's video lectures, so you can read and watch together.
LA-NET | ONLINE LEARNING VIDEO PLATFORM FOR POST-SECONDARY & HIGHER INSTITUTION STUDENTS
LA-NET | ONLINE LEARNING VIDEO PLATFORM FOR POST-SECONDARY & HIGHER INSTITUTION STUDENTS Series 1: Advanced Permutations, Combinations and Probability Laws Series outline • Part 1.1: Permutations • 1.1.1 Definition and properties of permutations • 1.1.2 Examples and applications of permutations • 1.1.3 Advanced counting with restrictions • Part 1.2: Combinations • 1.2.1 Definition and properties of combinations • 1.2.2 Examples and applications of combinations • 1.2.3 Distinction between permutations and combinations • Part 1.3: Probability Laws • 1.3.1 Definition of probability laws • 1.3.2 Addition and multiplication laws of probability • 1.3.3 Applications of probability laws in problem solving • Part 1.4: Applications of Permutations, Combinations and Probability Laws • 1.4.1 Real-life applications in probability modelling • 1.4.2 Problem-solving strategies using counting and probability laws • 1.4.3 Combined applications in discrete probability Introduction Probability is a powerful tool that helps us quantify uncertainty and make informed decisions in everyday life and scientific investigations. From arranging objects in specific orders to selecting subsets from larger collections, the concepts of permutations and combinations provide the foundation for counting methods in probability. These counting techniques, when combined with the fundamental laws
LA-NET | ONLINE LEARNING VIDEO PLATFORM FOR POST-SECONDARY & HIGHER INSTITUTION STUDENTS of probability, enable us to solve a wide range of problems in mathematics, statistics, and applied fields. This Series builds on your earlier knowledge of basic probability and introduces more advanced methods that will be essential for tackling complex problems later in the course. The aim of this Series is to equip you with the ability to define, explain, and apply advanced permutations, combinations, and probability laws in solving practical and theoretical problems. We shall commence this Series by exploring permutations, beginning with their definition, properties, and applications, including cases with restrictions. Next, we will move on to combinations, where we will examine their definition, properties, and applications, as well as the distinction between permutations and combinations. Following that, we will study probability laws, focusing on the addition and multiplication laws and their applications in problem solving. Finally, we will conclude this Series with applications of permutations, combinations, and probability laws, showing how these concepts are used together in real-life probability modelling and discrete probability problems. Series Learning Outcomes Upon completing this Series, you should be able to: • SLO 1.1 define permutations and explain their properties, • SLO 1.2 apply permutations to solve counting problems with and without restrictions, • SLO 1.3 define combinations and explain their properties, • SLO 1.4 distinguish between permutations and combinations with examples, • SLO 1.5 define probability laws and explain the addition and multiplication laws of probability, • SLO 1.6 apply probability laws to solve practical problems, and • SLO 1.7 evaluate combined applications of permutations, combinations, and probability laws in real-life probability modelling. 1.1 Permutations Permutations are a fundamental concept in probability and combinatorics. A permutation is an arrangement of objects in a specific order. The order of arrangement is important, which distinguishes permutations from combinations.
LA-NET | ONLINE LEARNING VIDEO PLATFORM FOR POST-SECONDARY & HIGHER INSTITUTION STUDENTS For example, arranging the letters A, B, and C as ABC, BAC, or CAB are all different permutations. Fill in the blank: A permutation is an arrangement of objects in a specific __________. 1.1.1 Definition and properties of permutations A permutation of ndistinct objects taken rat a time is defined as the number of ways of arranging robjects out of n. It is denoted by P(n,r)and given by the formula: P(n,r)=n!(n-r)! Here, factorial (denoted n!) means the product of all positive integers up to n. For example, 5!=5×4×3×2×1=120.
LA-NET | ONLINE LEARNING VIDEO PLATFORM FOR POST-SECONDARY & HIGHER INSTITUTION STUDENTS Illustration of permutations of three objects. 1.1.2 Examples and applications of permutations Let us consider a simple example. Suppose we want to know how many ways we can arrange the letters A, B, and C. Since there are three letters, the number of permutations is 3!=6. The arrangements are ABC, ACB, BAC, BCA, CAB, and CBA. Another example is arranging five students in a line. The number of possible arrangements is 5!=120. Permutations are widely applied in real life, such as in arranging seats, scheduling tasks, and designing codes. Fill in the blank: The number of ways to arrange three letters A, B, and C is equal to __________. Examples of permutations Scenario Number of permutations Arranging 3 letters 3!=6 Arranging 5 students 5!=120
LA-NET | ONLINE LEARNING VIDEO PLATFORM FOR POST-SECONDARY & HIGHER INSTITUTION STUDENTS Selecting 2 out of 4 objects P(4,2)=12 1.1.3 Advanced counting with restrictions Sometimes permutations involve restrictions. For example, arranging digits to form numbers where repetition is not allowed, or arranging people in seats where certain positions are fixed. Consider arranging the digits 1, 2, 3, and 4 to form a three-digit number without repetition. The first digit can be chosen in 4 ways, the second in 3 ways, and the third in 2 ways. Thus, the total number of permutations is 4×3×2=24. Fill in the blank: If four digits are used to form a three-digit number without repetition, the total number of permutations is __________. Key points on restricted permutations • Restrictions may include fixed positions or no repetition. • Multiply choices step by step to find total arrangements. • Useful in forming codes, passwords, and seating plans. Pilot questions 1.1 1. The formula for permutations of nobjects taken rat a time is: A. n!r!B. n!(n-r)!C. nrD. None of the above
LA-NET | ONLINE LEARNING VIDEO PLATFORM FOR POST-SECONDARY & HIGHER INSTITUTION STUDENTS 2. The number of ways to arrange three letters A, B, and C is: A. 3 B. 6 C. 9 D. 12 3. The factorial of 5, denoted 5!, is equal to: A. 24 B. 60 C. 120 D. 720 4. If four digits are used to form a three-digit number without repetition, the total number of permutations is: A. 12 B. 24 C. 36 D. 48 5. Which of the following scenarios involves restricted permutations? A. Arranging 5 students in a line B. Arranging digits to form a number without repetition C. Arranging letters in any order D. None of the above 1.2 Combinations Combinations are another key concept in probability and combinatorics. A combination is a selection of objects from a set where the order of selection does not matter. This is the main difference between combinations and permutations. For example, choosing the letters A, B, and C together is the same as choosing B, A, and C, since the arrangement is irrelevant. Fill in the blank: A combination is a selection of objects where __________ does not matter. 1.2.1 Definition and properties of combinations A combination of n distinct objects taken rat a time is denoted by C(n,r)or nr. The formula is: C(n,r)=n!r!(n-r)! This formula shows that combinations are derived from permutations but adjusted to remove the effect of order.
LA-NET | ONLINE LEARNING VIDEO PLATFORM FOR POST-SECONDARY & HIGHER INSTITUTION STUDENTS Illustration of combinations of three objects. 1.2.2 Examples and applications of combinations Suppose we want to know how many ways we can select two letters from A, B, and C. Using the formula: C(3,2)=3!2!(3-2)!=62=3 The possible selections are AB, AC, and BC. Another example is choosing a committee of 3 people from a group of 10. The number of possible committees is: C(10,3)=10!3!⋅7!=120 Combinations are widely applied in probability problems, such as lottery draws, team selections, and sampling. Fill in the blank: The number of ways to select two letters from A, B, and C is equal to __________. Examples of combinations Scenario Number of combinations Selecting 2 letters from 3 C(3,2)=3
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