Unit 3 - Quantitative Techniques (COQT111) PDF

Quantitative Techniques (COQT111) Unit 3 Foundation of Statistical Inference 1 Table of Contents Unit Overview....................................................................................................................... 3 1. B...

Quantitative Techniques (COQT111) Unit 3 Foundation of Statistical Inference 1 Table of Contents Unit Overview....................................................................................................................... 3 1. Basic Probability Concepts................................................................................... 4 2. Probability Distributions..................................................................................... 19 3. Unit Summary...................................................................................................... 24 4. References............................................................................................................ 24 2 Unit Overview Suppose a company questions 200 customers in order to estimate the proportion of all customers who favour a particular product. In this context, it would be expected that the proportion of the 200 customers in the survey in favour of the product is a representative of all customers who are in favour. There is a degree of uncertainty associated with any survey results. A determination of the likelihood that a certain proportion of the customers in the survey would favor the company product is of great importance in management-related decision-making. The task of calculating the likelihood that something occurs belongs to the realm of probability, which is the focus in this unit. Learning Outcomes By the end of this unit, you should be able to: Calculate probabilities using contingency tables. Calculate probabilities using both the Poisson and Binomial formulas, applying different techniques especially cumulative probabilities. 3 1. Basic Probability Concepts It is important to understand the terminologies commonly used when studying probability. The most common and basic terminologies relating to probability are defined below: Probability is defined as the likelihood (or chance) that a particular event will occur An experiment is a process by which an outcome is obtained (i.e., rolling a dice) The result of an experiment is called an outcome (i.e. obtaining a 2 after rolling a dice) An event is any particular outcome or group of outcomes (i.e. obtaining {Head}, or {Tail} or {Head; Tail} after tossing a fair coin) The sample space is the set of all possible outcomes of a random variable (i.e. after rolling a dice, all possible outcomes are given by {1; 2; 3; 4; 5; 6}) Mathematically, a probability is defined as the ratio of two numbers: 𝑟 𝑃(𝐴) = 𝑛 Where: 𝐴 = 𝑒𝑣𝑒𝑛𝑡 𝑜𝑓 𝑎 𝑠𝑝𝑒𝑐𝑖𝑓𝑖𝑐 𝑡𝑦𝑝𝑒 𝑟 = 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑜𝑢𝑡𝑐𝑜𝑚𝑒𝑠 𝑜𝑓 𝑒𝑣𝑒𝑛𝑡 𝐴 𝑛 = 𝑡𝑜𝑡𝑎𝑙 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑎𝑙𝑙 𝑝𝑜𝑠𝑠𝑖𝑏𝑙𝑒 𝑜𝑢𝑡𝑐𝑜𝑚𝑒𝑠 (𝑐𝑎𝑙𝑙𝑒𝑑 𝑠𝑎𝑚𝑝𝑙𝑒 𝑠𝑝𝑎𝑐𝑒) 𝑃(𝐴) = 𝑝𝑟𝑜𝑏𝑎𝑏𝑖𝑙𝑖𝑡𝑦 𝑜𝑓 𝑎𝑛 𝑒𝑣𝑒𝑛𝑡 𝐴 𝑜𝑐𝑐𝑢𝑟𝑖𝑛𝑔 Probability values are always defined on a scale from 0 to 1 (or an interval (0 ≤ 𝑃(𝐴) ≤ 1). A probability of zero (or near zero) indicates that an event is unlikely to occur, and a probability of 1 (or close to 1) indicates that an event is certain to occur. Other probability values between 0 and 1 represent degrees of likelihood that an event will occur. An illustration of the scale where the probability is defined is given in Figure 17. Figure 17: Probability scale (adapted from Anderson et al., 2011) 4 1.1. Types of Probability There are two types of probabilities namely: subjective and objective probabilities. Subjective probability is assigned based on personal feelings or insights. The determination of the probability comes from an educated guess, expert opinion or just plain intuition (Wegner, 2016). Although not a scientific approach to probability, the subjective method is often based on wisdom and experiences. Suppose an experienced head of the mathematics department at a college analyses students’ performance in mathematics at the end of term 1, and takes note of the poor performing students. Drawing form his experience, the head of department would have knowledge of the scope of the mathematics module and hence may be able to give an accurate probability that a certain proportion of the poor performing students would pass the final examination. However, this approach of determining probabilities is not used extensively in statistical analysis because it is difficult to statistically verify the correctness of the results. In contrast, objective probability employs scientific approaches to determining probabilities. Since scientific approaches are used, the probability of an event occurring can be verified statistically through surveys or empirical observations (Wegner, 2016). The objective probability approach is widely used in statistical analysis. 1.2. Properties of Probability When working with probabilities it is important to understand some of its most basic properties. A list of five most basic properties is as follows: A probability value always lies between 0 and 1 (i.e. 0 ≤ 𝑃(𝐴) ≤ 1). Note that 0 and 1 are included in the interval. If it is impossible for an event to occur, then 𝑃(𝐴) = 0. For example, the probability of a Spaza shop with capital investment of R8 000 making R80 000 profit in one day is zero. If it is certain that an event will occur, then 𝑃(𝐴) = 1. The probability that the human resource office at a company processes at least one leave application in a year is 1. The sum of the probabilities of all possible events equals 1 (i.e. for 𝑘 possible events in a sample space, 𝑃(𝐴1 ) + 𝑃(𝐴2 ) + 𝑃(𝐴3 ) + ⋯ + 𝑃(𝐴𝑘 ) = 1). For example if a coin is 1 1 tossed, there are two outcomes {Head, Tail), and 𝑃(𝐻𝑒𝑎𝑑) + 𝑃(𝑇𝑎𝑖𝑙) = + = 1. 2 2 Complementary probability: if 𝑃(𝐴) is the probability of event A occurring, then the probability of event A not occurring (i.e. 𝐴) is defined as 𝑃(𝐴) = 1 − 𝑃(𝐴). For example, if there is 60% chance that a salesperson would make R100 000,00 in one day then 60 𝑃(𝑚𝑎𝑘𝑖𝑛𝑔 𝑅100 000,00) = = 0,6 and 𝑃(𝑛𝑜𝑡 𝑚𝑎𝑘𝑖𝑛𝑔 𝑅100 000,00) = 1 − 0,6 = 0,4 100 Learn more about this 5  More information relating to types of probability, see Wegner (2016, p.107-109) 1.3. Probability Concepts It is important to understand some basic probability concepts so that working with problematic situations involving probabilities becomes easier. In this section, we focus on the following basic probability concepts. The intersection of events The union of events Mutually exclusive events Collectively exhaustive events Statistically independent events The following example will help us to understand the differences between these concepts. Example 12: Consider the table below showing recruitment by sex at a new company. Department Sex Total Males Females Transport 3 3 6 Marketing 3 5 8 Security 6 4 10 Human resources 3 1 4 Total 15 13 28 Table 5: Cross-tabulation table - recruitment by sex a) Intersection of events The intersection of two events 𝐴 and 𝐵 is the set of all outcomes that belong to both 𝐴 and 𝐵 simultaneously. It is written as 𝐴 ∩ 𝐵 (i.e. 𝐴 and B). 6  Figure 18: Venn diagram showing intersection of two events (𝐴 ∩ 𝐵) To illustrate, we answer the following question: What is the probability that a randomly selected employee will be female and belong to the security department? Solution: Let 𝐴 = event (female employee) Let 𝐵 = event (security employee) Then (𝐴 ∩ 𝐵) the set of all employees who are female ‘and’ are recruited in security department. From table 5, there are 4 employees out of 28 who are female and recruited in security department. 4 Thus 𝑃(𝐴 ∩ 𝐵) = 𝑃(𝐹𝑒𝑚𝑎𝑙𝑒 ∩ 𝑆𝑒𝑐𝑢𝑟𝑖𝑡𝑦) = 28 = 0,143 A graphical representation is shown in Figure 19: 7  Figure 19: Venn diagram of female and security employees b) Union of events The union of two events 𝐴 and 𝐵 is the set of all outcomes that belong to either 𝐴 or 𝐵 or both. It is written as 𝐴 ∪ 𝐵. Figure 20: Venn diagram of the union of events (𝐴 ∪ 𝐵) 8 To illustrate, we answer the following question: What is the probability that a randomly selected employee will be female or will be an employee in the security department, or both? Solution: Let 𝐴 = event (female employee) Let 𝐵 = event (security employee) Then (𝐴 ∪ 𝐵) the set of all employees who are female or employed in security department or both (female and security) employees. From table 5, there are 13 female employees (includes 4 security employees), 10 security employees (includes 4 female employees) and 4 employees who are female and in security department. This means that there are 19 different employees (13 + 10 − 4) that are either female or in security or both. 13+10−4 19 Thus 𝑃(𝐴 ∪ 𝐵) = 𝑃(𝑓𝑒𝑚𝑎𝑙𝑒 ∪ 𝑆𝑒𝑐𝑢𝑟𝑖𝑡𝑦) = 28 = 28 = 0,679 A graphical representation is shown in Figure 21: Figure 21: Venn diagram of white or security employees 9 c) Mutually exclusive Events are mutually exclusive if they cannot occur together on a single trial of a random experiment (i.e. not at the same point in time). Figure 22: Venn diagram showing mutually exclusive events 𝐴 ∩ 𝐵 = 0 For example, we answer the question: what is the probability of randomly selecting an employee who is both male and female? Solution: Let 𝐴 = event (male employee) Let 𝐵 = event (female employee) Events A and B are mutually exclusive because a randomly selected employee cannot be male and female at the same time 0 Thus 𝑃(𝐴 ∩ 𝐵) = 𝑃(𝑚𝑎𝑙𝑒 ∩ 𝑓𝑒𝑚𝑎𝑙𝑒) = 28 = 0. A probability of zero means an impossible event. d) Collectively exhaustive events 10 Events are collectively exhaustive when the union of all possible events is equal to the sample space. To illustrate we answer the question: what is the probability of selecting an employee who is male or female from the sample of 28 employees (refer to table 5). Solution: Let 𝐴 = event (male employee) Let 𝐵 = event (female employee) And we have (𝐴 ∪ 𝐵) as the sample space of all employees 15 13 Thus 𝑃(𝐴 ∪ 𝐵) = 𝑃(𝑚𝑎𝑙𝑒) + 𝑃(𝑓𝑒𝑚𝑎𝑙𝑒) = + =1 28 28 e) Statistically independent events Two events 𝐴 and 𝐵 are statistically independent if the occurrence of event 𝐴 has no effect on the outcome of event 𝐵 and vice-versa. It is also important to note the key difference between ‘mutually exclusive events’ and ‘statistically independent events’ to avoid confusion. When two events are mutually exclusive, they cannot occur together. On the other hand, when two events are statistically independent, they can occur together, but they do not have an influence on each other. Here is an example for statistically independent events: Let A = event (The manager will have a meeting in the boardroom) Let B = event (It will rain) Notice here that the occurrence of the meeting in the boardroom does not influence the weather and vice-versa. In fact, the two events can occur simultaneously. Learn more about this More information relating to basic probability concepts, see Wegner (2016, p.109 -113) Calculating Objective Probabilities Objective probabilities can be classified into three types namely: marginal probability, joint probability, and conditional probability. We illustrate their differences using an example. Example 13: Consider the information relating to numbers of people (grouped according to gender) who have a particular blood group. 11  Table 6: Blood groups Marginal probability P(A): a marginal probability is a probability of a single event ‘A’ occurring only. A frequency table is used to find marginal probabilities because it shows the outcomes of only one random variable (Wegner, 2016). It is named ‘marginal’ because computing probability uses values on the margins of a frequency 𝑁𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑝𝑒𝑜𝑝𝑙𝑒 𝑤𝑖𝑡ℎ 𝑏𝑙𝑜𝑜𝑑 𝑔𝑟𝑜𝑢𝑝 𝐴𝐵 22 table. In example 13, 𝑃(𝐵𝑙𝑜𝑜𝑑 𝑔𝑟𝑜𝑢𝑝 𝐴𝐵) = 𝑇𝑜𝑡𝑎𝑙 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑝𝑒𝑜𝑝𝑙𝑒 = 140 = 0,16. Joint probability P(𝐴 ∩ 𝐵): a joint probability is the probability that both event A and event B will occur simultaneously on a single trial of a random experiment (Wegner, 2016). A joint event refers to the outcomes of two or more random variables occurring together. It is the same as the intersection of two events in a Venn diagram. A cross tabulation is used to find joint probabilities because it shows the outcomes of two random variables. In example 13, the probability of being female and belong to blood group O can be calculated as follows: 27 𝑃(𝐹𝑒𝑚𝑎𝑙𝑒 𝑎𝑛𝑑 𝑂) = 𝑃(𝐹 ∩ 𝑂) = = 0,19 140 Conditional probability P(𝐴|𝐵): a conditional probability is the probability of event A occurring, given that event B has already occurred. It is the probability of an event on condition that a certain criteria is satisfied. It is written as P(𝐴|𝐵). The formula is: 𝑃(𝐴 ∩ 𝐵) 𝑃(𝐵) = 𝑃(𝐵) The key feature of conditional probability is that the sample space is reduced to the set of outcomes associated with the ‘given’ prior to the occurrence of event B only. The prior information (i.e. event B) can change the likelihood of event A occurring. In example 13, if an individual was selected randomly and found to be male, what is the probability that he has a blood group A? Here the total possible outcomes constitute a subset (males i.e. 70) of the total number of people. This probability is read as probability of A given M. 12  𝑁𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑚𝑎𝑙𝑒𝑠 𝑤ℎ𝑜 ℎ𝑎𝑠 𝑎 𝑏𝑙𝑜𝑜𝑑 𝑔𝑟𝑜𝑢𝑝 𝑂 21 𝑃(𝑀) = = 𝑇𝑜𝑡𝑎𝑙 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑚𝑎𝑙𝑒𝑠 70 Learn more about this For more information relating to calculating objective probabilities, see Wegner (2016). 1.4. Probability Rules The probability rules are more useful when calculating probabilities of compound or multiple events occurring simultaneously. Two probability rules are presented below: The addition rule: useful for non-mutually exclusive events and mutually exclusive events. It relates to the union of events. Used to find the probability of either event A or event B, or both events occurring simultaneously in a single trial of a random experiment The multiplication rule: useful for statistically dependent events and statistically independent events. It relates to the union of events. It is used to find the probability of event A and event B occurring together in a single trial of a random experiment. Additional rule: non-mutually exclusive events Non-mutually exclusive events are those events that can occur together in a single trial of a random experiment. Therefore the probability of either event A or event B or both occurring in a single trial of a random experiment is defined as: 𝑃(𝐴 ∪ 𝐵) = 𝑃(𝐴) + 𝑃(𝐵) − 𝑃(𝐴 ∩ 𝐵) If a Venn diagram is used for illustration, the union of two non-mutually exclusive events is the combined outcomes of the two overlapping events A and B. From example 13, if an individual was selected randomly, what is the probability that the individual is either female or someone with blood group AB, or both? Solution: Let 𝐴 = event (Female) Let 𝐵 = event (Blood group AB) Notice that the two events are not mutually exclusive as they occur at the same time. 13  70 Therefore: 𝑃(𝐴) = 𝑃(𝐹𝑒𝑚𝑎𝑙𝑒) = 140 = 0,5 22 𝑃(𝐵) = 𝑃(𝐴𝐵) = 140 = 0,157 10 𝑃(𝐴 ∩ 𝐵) = 𝑃(𝐹𝑒𝑚𝑎𝑙𝑒 𝑤𝑖𝑡ℎ 𝑏𝑙𝑜𝑜𝑑 𝑔𝑟𝑜𝑢𝑝 𝐴𝐵) = = 0,0714 140 Then 𝑃(𝐴 ∪ 𝐵) = 𝑃(𝑒𝑖𝑡ℎ𝑒𝑟 𝑓𝑒𝑚𝑎𝑙𝑒 𝑜𝑟 𝑔𝑟𝑜𝑢𝑝 𝐴𝐵 𝑜𝑟 𝑏𝑜𝑡ℎ) = 𝑃(𝐹𝑒𝑚𝑎𝑙𝑒) + 𝑃(𝐺𝑟𝑜𝑢𝑝 𝐴𝐵) − 𝑃(𝐹𝑒𝑚𝑎𝑙𝑒 𝑤𝑖𝑡ℎ 𝑏𝑙𝑜𝑜𝑑 𝑔𝑟𝑜𝑢𝑝 𝐴𝐵) = 0,5 + 0,157 − 0,0714 = 0,5856 Additional rule: mutually exclusive events Mutually exclusive events are those events that cannot occur together in a single trial of a random experiment. For mutually exclusive events, there is no intersectional event, meaning that 𝑃(𝐴 ∩ 𝐵) = 0. Thus the probability of either event A or event B (but not both) occurring in a single trial of a random experiment is defined as: 𝑃(𝐴 ∪ 𝐵) = 𝑃(𝐴) + 𝑃(𝐵) If events are mutually exclusive, then the probability is the sum of only the two marginal probabilities of events A and B. Using Venn diagrams, the union of two mutually exclusive events is the sum of the outcomes of each of the two (non-overlapping) events A and B separately. For example; what is the probability that a randomly selected individual is either someone with blood group O or AB? Solution: Let 𝐴 = event (person with blood group O) Let 𝐵 = event (person with blood group AB) The two events are mutually exclusive since they cannot occur simultaneously. Therefore 𝑃(𝐴 ∩ 𝐵) = 0. 50 Since: 𝑃(𝐴) = 𝑃(𝐺𝑟𝑜𝑢𝑝 𝑂) = 140 = 0,357 22 𝑃(𝐵) = 𝑃(𝐺𝑟𝑜𝑢𝑝 𝐴𝐵) = 140 = 0,157 Then 𝑃(𝐴 ∪ 𝐵) = 𝑃(𝐺𝑟𝑜𝑢𝑝 𝑂) + 𝑃(𝐺𝑟𝑜𝑢𝑝 𝐴𝐵) = 0,357 + 0,157 = 0,514 Multiplication Rule for statistically dependent events The multiplication rule is often used to find the joint probability of events A and B occurring together in a single trial of random experiment (i.e. the intersection of the two events). This 14 rule assumes that the two events A and B are associated (i.e. they are dependent events). The multiplication rule for dependent events is given by the following: 𝑃(𝐴 ∩ 𝐵) = 𝑃(𝐴|𝐵) × 𝑃(𝐵) Where: 𝑃(𝐴 ∩ 𝐵) = 𝑗𝑜𝑖𝑛𝑡 𝑝𝑟𝑜𝑏𝑎𝑏𝑖𝑙𝑖𝑡𝑦 𝑜𝑓 𝐴 𝑎𝑛𝑑 𝐵 𝑃(𝐵) = 𝑐𝑜𝑛𝑑𝑖𝑡𝑖𝑜𝑛𝑎𝑙 𝑝𝑟𝑜𝑏𝑎𝑏𝑖𝑙𝑖𝑡𝑦 𝑜𝑓 𝐴 𝑔𝑖𝑣𝑒𝑛 𝐵 𝑃(𝐵) = 𝑚𝑎𝑟𝑔𝑖𝑛𝑎𝑙 𝑝𝑟𝑜𝑏𝑎𝑏𝑖𝑙𝑖𝑡𝑦 𝑜𝑓 𝐵 𝑜𝑛𝑙𝑦 Example 14 Suppose there are 3 green marbles, 4 blue marbles, and 3 red marbles in a bag. What is the probability of drawing 2 blue marbles from the bag if the first marble is not replaced before the second marble is drawn? Solution: Let A = event (drawing first blue marble) Let B = event (drawing second blue marble) These two events are dependent because the second event is influenced by the first event (i.e. the marble drawn at first is not replaced). There are 10 marbles altogether in the bag. 4 𝑃(1𝑠𝑡 𝑏𝑙𝑢𝑒 𝑚𝑎𝑟𝑏𝑙𝑒) = 10 Since the first marble drawn is not replaced, then there are 9 marbles left in the bag altogether. 3 Of these 9 marbles left in the bag, only 3 are blue. So 𝑃(2𝑛𝑑 𝑏𝑙𝑢𝑒 𝑚𝑎𝑟𝑏𝑙𝑒) = 9 Therefore, 4 3 2 𝑃(1𝑠𝑡 𝑏𝑙𝑢𝑒 𝑚𝑎𝑟𝑏𝑙𝑒) × 𝑃(1𝑠𝑡 𝑏𝑙𝑢𝑒 𝑚𝑎𝑟𝑏𝑙𝑒) = × = 10 9 15 Multiplication Rule for statistically independent events If two events, A and B, are statistically independent (i.e. there is no association between the two events) then the multiplication rule reduces to the product of the two marginal probabilities only: 𝑃(𝐴 ∩ 𝐵) = 𝑃(𝐴) × 𝑃(𝐵) Where: 𝑃(𝐴 ∩ 𝐵) = 𝑗𝑜𝑖𝑛𝑡 𝑝𝑟𝑜𝑏𝑎𝑏𝑖𝑙𝑖𝑡𝑦 𝑜𝑓 𝐴 𝑎𝑛𝑑 𝐵 𝑃(𝐴) = 𝑚𝑎𝑟𝑔𝑖𝑛𝑎𝑙 𝑝𝑟𝑜𝑏𝑎𝑏𝑖𝑙𝑖𝑡𝑦 𝑜𝑓 𝐴 𝑜𝑛𝑙𝑦 𝑃(𝐵) = 𝑚𝑎𝑟𝑔𝑖𝑛𝑎𝑙 𝑝𝑟𝑜𝑏𝑎𝑏𝑖𝑙𝑖𝑡𝑦 𝑜𝑓 𝐵 𝑜𝑛𝑙𝑦 The following test is used to check if two events are statistically independent. Two events are statistically independent if the following is true: 𝑃(𝐵) = 𝑃(𝐴) This means that the prior occurrence of event B does not influence the outcome of event A. 15 Example 15 Suppose you throw two fair dice. What is the probability of obtaining a 3 on each dice? Solution: The two events are independent – the outcome of the first event (throwing the first die) does not influence the occurrence of the second event (throwing the second die). 𝑃(3 𝑜𝑛 1𝑠𝑡 𝑑𝑖𝑒 𝑎𝑛𝑑 3 𝑜𝑛 2𝑛𝑑 𝑑𝑖𝑒) = 𝑃(3 𝑜𝑛 𝑓𝑖𝑟𝑠𝑡 𝑑𝑖𝑒) × 𝑃(3 𝑜𝑛 𝑠𝑒𝑐𝑜𝑛𝑑 𝑑𝑖𝑒) 1 1 = × 6 6 1 = 36 Learn more about this For more information relating to probability rules, see Wegner (2016) 1.5. Probability Trees A probability tree is a graphic way to apply probability rules where there are multiple events that occur in sequence and these events can be represented by branches (similar to a tree). Example 16: Consider two fair coins that are tossed, find the outcomes using a tree diagram. What is the probability that you will obtain two heads? Solution: 16 The outcomes are {(H,H); (H,T); (T,H); (T,T)} 1 Probability 𝑃(𝑇𝑤𝑜 𝐻𝑒𝑎𝑑𝑠) = 𝑃(𝐻, 𝐻) = 4 Learn more about this Solving probabilities using tree diagrams can be very challenging. This video provides more detailed and clear explanations: https://www.youtube.com/watch?v=kNOrDWm15bY https://www.youtube.com/watch?v=z9FKoQ8a_1E 1.6. The Counting Rules: Permutations and Combinations Probability calculations involve counting the number of event outcomes (𝑟) and the total number of possible outcomes (𝑛) and expressing this as a ratio. Often the values for 𝑟 and 𝑛 cannot be counted because of the large number of possible outcomes involved. Counting rules help to find values for 𝑟 and 𝑛. There are three basic counting rules namely: the multiplication rule, the permutation rule, and the combination rule (Wegner, 2016). a) Multiplication rule of counting For a single event, the total number of different ways (unique ways) in which 𝑛 objects (i.e. the full sample space) can be arranged (ordered) is given by 𝑛! (read as ‘n factorial’). 𝑛! = 𝑛 × (𝑛 − 1) × (𝑛 − 2) × (𝑛 − 3) × … × 3 × 2 × 1 (note that 0! = 1) For example, consider the number of unique arrangements of seven 100m athletes in a seven- lane track. The number of different arrangements of the seven athletes will be: 7! = 7 × 6 × 5 × 4 × 3 × 2 × 1 = 5040 17 For combined events, if a random process has 𝑛1 possible outcomes for event 1, 𝑛2 possible outcomes for event 2, …, 𝑛𝑗 possible outcomes for event 𝑗, then the total number of possible outcomes for the 𝑗 events collectively is: 𝑛1 × 𝑛2 × 𝑛3 × … × 𝑛𝑗 b) Permutation rule of counting A permutation is a number of distinct (different) ways of selecting (or arranging) a subset of 𝑟objects drawn from a larger group of 𝑛 objects, where the order of selecting objects is important. Each possible arrangement of the subset of 𝑟 objects is called a permutation. The number of different ways of arranging 𝑟 objects selected from 𝑛 objects, where the order is important is given by: 𝑛! 𝑛 𝑃𝑟 = (𝑛 − 𝑟)! Where: 𝑟 = 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑜𝑏𝑗𝑒𝑐𝑡𝑠 𝑠𝑒𝑙𝑒𝑐𝑡𝑒𝑑 𝑎𝑡 𝑎 𝑡𝑖𝑚𝑒 𝑛 = 𝑡𝑜𝑡𝑎𝑙 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑜𝑏𝑗𝑒𝑐𝑡𝑠 𝑓𝑟𝑜𝑚 𝑤ℎ𝑖𝑐ℎ 𝑡𝑜 𝑠𝑒𝑙𝑒𝑐𝑡 Example 17: A Statistics debating team consists of 5 speakers. a) In how many ways can all 5 speakers be arranged in a row for a photo? b) How many ways can the captain and vice-captain be chosen? Solution: a) The speakers can be arranged in 120 ways 5! 5! 5×4×3×2×1 5! = 5 × 4 × 3 × 2 × 1 = 120 OR 5 𝑃5 = (5−5)! = = = 120 0! 1 b) Two people can be chosen, captain and vice-captain 5! 5! 5×4×3×2×1 120 Therefore 5 × 4 = 20 OR 5 𝑃2 = (5−2)! = 3! = 3×2×1 = 6 =2 c) Combination rule A combination is the number of distinct ways of selecting (or arranging) a subset of 𝑟 objects drawn from a larger group of 𝑛 objects where the order of selecting objects is not important. Each separate grouping of the subset of 𝑟 objects is called a combination. The number of ways of selecting 𝑟 objects selected from 𝑛 objects, not considering the order of selection is given by: 𝑛! 𝑛 𝐶𝑟 = 𝑟! (𝑛 − 𝑟)! Where: 𝑟 = 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑜𝑏𝑗𝑒𝑐𝑡𝑠 𝑠𝑒𝑙𝑒𝑐𝑡𝑒𝑑 𝑎𝑡 𝑎 𝑡𝑖𝑚𝑒 𝑛 = 𝑡𝑜𝑡𝑎𝑙 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑜𝑏𝑗𝑒𝑐𝑡𝑠 𝑓𝑟𝑜𝑚 𝑤ℎ𝑖𝑐ℎ 𝑡𝑜 𝑠𝑒𝑙𝑒𝑐𝑡 18 Example 18: How many ways can a basketball team of 5 players be chosen from 9 players? Solution: 9! 9! 362880 9 𝐶5 = = = = 126 5! (9 − 5)! 5! 4! (120)(24) Learn more about this For more information regarding permutations and combinations, go to this link https://www.youtube.com/watch?v=XPPYYM6WCuE This video provides information that summarizes basic probability concepts https://www.youtube.com/watch?v=CBnGs9t6RxY 2. Probability Distributions A probability distribution is a list of all the possible outcomes of a random variable and their associated probabilities of occurrence (Wegner, 2016). The probability distribution for a random variable X provides the possible outcomes for X, and the probabilities associated with each possible value. 2.1. Types of Probability Distributions Probability distributions functions can be classified as either discrete or continuous. While a discrete random variable takes on a finite or countably infinite number of distinct possible values, a continuous random variable takes on an infinite number of possible values. In this section, the focus is on two discrete probability functions (called the binomial probability distribution, and the Poisson probability distribution) and one continuous probability function (called the normal probability distribution). Learn more about this For more information relating to types of probability distributions, see Wegner (2016, p.133) 19 2.2. Discrete Probability Distributions Discrete probability distributions assume that the outcomes of a random variable under study can take on only specific values (normally integers). These distributions assign a probability to each value of a discrete random variable X. Some examples of discrete probability distributions are: Number of enrolled Engineering students (i.e. 0, 1, 2, 3, …) Number of faulty machines at a company (i.e. 0, 1, 2, 3, …) Number of people using the Gautrain per day (i.e. 0, 1, 2, 3, …) For each possible outcome of a discrete random variable in a sample space, there is a non- zero probability. The zero probability is only assigned for values of the random variable outside the sample space. As noted above, binomial probability distribution and Poisson probability distribution are the commonly used discrete probability distribution functions. Learn more about this More information and detailed explanations about discrete probability distributions, please click the on the link below: https://www.youtube.com/watch?v=UnzbuqgU2LE 2.3. Binomial Probability Distribution A discrete random variable follows the binomial distribution if it satisfies the following four conditions: The random variable is observed 𝑛 number of times (𝑛 = 1, 2, 3, …) There are only two, mutually exclusive and collectively exhaustive, outcomes associated with the random variable on each object in the sample. These two outcomes are labelled ‘success’ and ‘failure’ (i.e. an application is successful or unsuccessful, a motor vehicle is insured or not insured). Each outcome has an associated probability. The probability for the success outcome is denoted by 𝑝, and the probability for the failure outcome is denoted by 1 − 𝑝. The objects are assumed to be independent of each other, meaning the 𝑝 remains constant for each sampled object. Before we calculate the probabilities for probability distributions, let us first discuss the meanings of the following phrases that are often used in probability problem solving. 20 At least three (𝑥 ≥ 3): This means that three is the minimum value and if we say at least three books, it means three or four or five books. At most three (𝑋 ≤ 3): This means that three is the maximum value. At most three books means no book or one book or two books or three books. No more than three (𝑋 ≤ 3): This means that three is the maximum number. With regards to number of books, we would say that it means, three or two or one or zero books. Less than three (𝑋 < 3): This means that three is not included and we are only interested in the values smaller than three, that is, zero or one or two. More than three 𝑋 > 3): This means that three is not included and we are only interested in the values larger than three, that is, four, five, six etc. Once these four conditions are satisfied, the following binomial question can be answered. Binomial question What is the probability that 𝑥 successes will occur in a randomly drawn sample of 𝑛 objects? The following formula is used to address the binomial question: 𝑃(𝑥) = 𝑛 𝐶𝑥 𝑝 𝑥 (1 − 𝑝)𝑛−𝑥 𝑓𝑜𝑟 𝑥 = 0, 1, 2, 3, … , 𝑛 Where: 𝑛 = 𝑡ℎ𝑒 𝑠𝑎𝑚𝑝𝑙𝑒 𝑠𝑖𝑧𝑒 𝑖. 𝑒. 𝑡ℎ𝑒 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑖𝑛𝑑𝑒𝑝𝑒𝑛𝑑𝑒𝑛𝑡 𝑜𝑏𝑠𝑒𝑟𝑣𝑎𝑡𝑖𝑜𝑛𝑠 𝑥 = 𝑡ℎ𝑒 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑠𝑢𝑐𝑐𝑒𝑠𝑠 𝑜𝑢𝑡𝑐𝑜𝑚𝑒𝑠 𝑖𝑛 𝑡ℎ𝑒 ′𝑛′ 𝑖𝑛𝑑𝑒𝑝𝑒𝑛𝑑𝑒𝑛𝑡𝑙𝑦 𝑑𝑟𝑎𝑤𝑛 𝑜𝑏𝑗𝑒𝑐𝑡𝑠 𝑝 = 𝑝𝑟𝑜𝑏𝑎𝑏𝑖𝑙𝑖𝑡𝑦 𝑜𝑓 𝑎 𝑠𝑢𝑐𝑐𝑒𝑠𝑠 𝑜𝑢𝑡𝑐𝑜𝑚𝑒 𝑜𝑛 𝑎 𝑠𝑖𝑛𝑔𝑙𝑒 𝑖𝑛𝑑𝑒𝑝𝑒𝑛𝑑𝑒𝑛𝑡 𝑜𝑏𝑗𝑒𝑐𝑡 (1 − 𝑝) = 𝑝𝑟𝑜𝑏𝑎𝑏𝑖𝑙𝑖𝑡𝑦 𝑜𝑓 𝑎 𝑓𝑎𝑖𝑙𝑢𝑟𝑒 𝑜𝑢𝑡𝑐𝑜𝑚𝑒 𝑜𝑛 𝑎 𝑠𝑖𝑛𝑔𝑙𝑒 𝑖𝑛𝑑𝑒𝑝𝑒𝑛𝑑𝑒𝑛𝑡 𝑜𝑏𝑗𝑒𝑐𝑡 The 𝑥 values (called the domain) represent the number of success outcome that can be observed in a sample of 𝑛 objects. It is also important to note that the success outcome is always associated with the probability, 𝑝. Thus the outcome that must be labelled as the success outcome is identified from the binomial question. Example 19: A surgery has a success rate of 85%. Suppose that the surgery is performed on three patients. What is the probability that the surgery is successful on exactly 2 patients? Solution: The number of successful surgeries, 𝑋 can be represented by a binomial distribution with 𝑛 = 3 trials, success probability 𝑝 = 0,85 and failure probability 𝑞 = 1 − 𝑝 = 0,15. Therefore: 𝑃(2) = 3 𝐶2 𝑝2 𝑞 𝑛−2 = 3 𝐶2 (0,85)2 (0,15)1 = 0,325 Descriptive statistical measures of the Binomial distribution 21 A measure of central location and a measure of dispersion can be calculated for any random variable that follows a binomial distribution using the following formulae: 𝑀𝑒𝑎𝑛: 𝜇 = 𝑛𝑝 𝑆𝑡𝑎𝑛𝑑𝑎𝑟𝑑 𝑑𝑒𝑣𝑖𝑎𝑡𝑖𝑜𝑛: 𝜎 = √𝑛𝑝(1 − 𝑝) From Example 19, calculate the mean of 𝑋. Solution: 𝑀𝑒𝑎𝑛: 𝜇 = 3 × 0,85 = 2,55 Learn more about this For more information relating to binomial probability distributions, see Wegner (2016) 2.4. Poisson Probability Distribution As noted above, a Poisson probability distribution is a discrete process. A Poisson process measures the number of occurrences of a particular outcome of a discrete random variable in a predetermined interval, that is, interval of time, space, and volume, for which an average number of occurrences of the outcome is known or can be determined (Wegner, 2016). The conditions satisfied by a Poisson distribution are as follows: The occurrence or non-occurrence of the event over any interval is independent of the occurrence or non-occurrence over any other interval. The probability of the occurrence of an event is the same for any two intervals of the same length. The occurrences are uniformly distributed throughout the interval of time Some of the examples of a Poisson process (also see Wegner, 2016) are: The number of breakdowns of a machine in a day The number of sales made by a salesperson in one month The number of challenges identified at the end of a field project The hourly number of customers arriving at a bank The number of typos in a book From these examples, it can be seen that the number of occurrences of a given outcome of the random variable, 𝑥, can take on any integer value from 0 to infinity (i.e. 0 ≤ 𝑥 < ∞). 22 Poisson question What is the probability of 𝑥 occurrences of a given outcome being observed in a predetermined time, space or volume interval? To answer the Poisson question, the following Poisson probability distribution formula can be used: 𝑒 −𝜆 𝜆𝑥 𝑃(𝑥) = 𝑓𝑜𝑟 𝑥 = 0, 1, 2, 3, … 𝑥! Where: 𝜆= 𝑡ℎ𝑒 𝑚𝑒𝑎𝑛 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑜𝑐𝑐𝑢𝑟𝑒𝑛𝑐𝑒𝑠 𝑜𝑓 𝑎 𝑔𝑖𝑣𝑒𝑛 𝑜𝑢𝑡𝑐𝑜𝑚𝑒 𝑜𝑓 𝑡ℎ𝑒 𝑟𝑎𝑛𝑑𝑜𝑚 𝑣𝑎𝑟𝑖𝑎𝑏𝑙𝑒 𝑓𝑜𝑟 𝑎 𝑝𝑟𝑒𝑑𝑒𝑡𝑒𝑟𝑚𝑖𝑛𝑒𝑑 𝑡𝑖𝑚𝑒, 𝑠𝑝𝑎𝑐𝑒 𝑜𝑟 𝑣𝑜𝑙𝑢𝑚𝑒 𝑖𝑛𝑡𝑒𝑟𝑣𝑎𝑙 𝑒 = 𝑎 𝑚𝑎𝑡ℎ𝑒𝑚𝑎𝑡𝑖𝑐𝑎𝑙 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡, 𝑎𝑝𝑝𝑟𝑜𝑥𝑖𝑚𝑎𝑡𝑒𝑙𝑦 𝑒𝑞𝑢𝑎𝑙 𝑡𝑜 2,71828 𝑥= 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑜𝑐𝑐𝑢𝑟𝑒𝑛𝑐𝑒𝑠 𝑜𝑓 𝑎 𝑔𝑖𝑣𝑒𝑛 𝑜𝑢𝑡𝑐𝑜𝑚𝑒 𝑓𝑜𝑟 𝑤ℎ𝑖𝑐ℎ 𝑎 𝑝𝑟𝑜𝑏𝑎𝑏𝑖𝑙𝑖𝑡𝑦 𝑖𝑠 𝑟𝑒𝑞𝑢𝑖𝑟𝑒𝑑 (𝑥 𝑐𝑎𝑛 𝑏𝑒 𝑎𝑛𝑦 𝑑𝑖𝑠𝑐𝑟𝑒𝑡𝑒 𝑣𝑎𝑙𝑢𝑒 𝑓𝑟𝑜𝑚 0 𝑡𝑜 𝑖𝑛𝑓𝑖𝑛𝑖𝑡𝑦) Example 20: The number of typographical errors in a PhD thesis is Poisson distributed with a mean of 1,9 per 100 pages. If 100 pages of the thesis are seleceted randomly, what is the probability that there are no typographical errors? 𝑒 −𝜆 𝜆𝑥 𝑒 −1,9 1,90 𝑒 −1,9 (1) Solution: 𝑃(𝑋 = 0) = 𝑥! = 0! = 1 = 0,1496 Descriptive statistical measures of the Poisson distribution A measure of central location and dispersion can be calculated for any random variable that follows a Poisson process using the following formula: 𝑀𝑒𝑎𝑛: 𝜇 = 𝜆 𝑆𝑡𝑎𝑛𝑑𝑎𝑟𝑑 𝑑𝑒𝑣𝑖𝑎𝑡𝑖𝑜𝑛: 𝜎 = √𝜆 Learn more about this For more information about binomial and Poisson probability distributions, see this video https://www.youtube.com/watch?v=BR1nN8DW2Vg 23  For more information relating to binomial probability distributions, see Wegner (2016: 141-150) 3. Unit Summary This unit provided an introduction to probability concepts, probability distribution, and sampling. Specifically, the focus was on 1) Basic Probability Concepts (Types of Probability, Properties of Probability, Probability Concepts, Objective Probabilities, Probability Rules, Probability Trees, Counting Rules: Permutations and Combinations), 2) Probability Distribution (Types of Probability Distributions, Discrete Probability Distributions, Binomial Probability Distribution, Poisson Probability Distribution, Continuous Probability Distributions, Normal Probability Distribution. The following video provides further insights in terms of understanding probability distribution: https://www.youtube.com/watch?v=UnzbuqgU2LE&t=1368s 4. References Anderson, D., Sweeney, D., Williams, T. (2011) Statistics for Business and Economics, South- Western, Cengage Learning Doane, D., & Seward, L. (2011). Applied Statistics in business and Economics (3rd Ed). Mcgraw-Hill Groebner, D., Shannon, P., Fry, P., & Smith, K. (2011). Business Statistics: a decision-making approach. Pearson Education Inc. Levine, D., Krehbiel, T. & Berenson, M. (2009). Business Statistics: A First Course, (5th Edition), Prentice-Hall, Inc. Wegner, T. (2016). Applied Business Statistics: Methods and Excel-based applications (4th Ed). Juta & Company 24