Statistics Answer Note #4

Question 1 Jennifer has recently interviewed for two different jobs. She feels there is a 0.380 probability of being offered the first job and a 0.560 probability of being offered the second job. Assume that the two job offers are statistically independent. a. What is the probability that Jennifer will be offered both jobs? Do not round intermediate calculations. Round your answer to three decimal places. Probability = b. What is the probability that Jennifer will be offered neither of those two jobs? Do not round intermediate calculations. Round your answer to three decimal places. Probability = c. What is the probability that Jennifer will be offered at least one of the two jobs? Do not round intermediate calculations. Round your answer to three decimal places. Probability = d. What is the probability that Jennifer will be offered the first job but not the second job? Do not round intermediate calculations. Round your answer to three decimal places. Probability = e. What is the probability the Jennifer will not be offered the first job but will be offered the second job? Do not round intermediate calculations. Round your answer to three decimal places. Probability = Answer The probability of Jennifer being offered both jobs is calculated by multiplying the probabilities of being offered each job together, since the events are independent. Probability of being offered first job = 0.380 Probability of being offered second job = 0.560 Therefore, P(both jobs) = 0.380 x 0.560 = 0.2128 Rounded to three decimal places, the probability that Jennifer will be offered both jobs is 0.213. b. What is the probability that Jennifer will be offered neither of those two jobs? The probability that Jennifer will be offered neither of the two jobs can be calculated by subtracting the probability of being offered one job from 1. Probability of being offered at least one job = 1 - [P(both jobs)] = 1 - 0.2128 = 0.7872 Rounded to three decimal places, the probability that Jennifer will be offered neither of the two jobs is 0.787. c. What is the probability that Jennifer will be offered at least one of the two jobs? The probability of Jennifer being offered at least one of the two jobs can be calculated by adding the probabilities of being offered each job separately and then subtracting the probability of both jobs together, as this will avoid double counting. Probability of being offered first job = 0.380 Probability of being offered second job = 0.560 Probability of both jobs = 0.2128 Therefore, P(at least one job) = (0.380 + 0.560) - 0.2128 = 0.7272 Rounded to three decimal places, the probability that Jennifer will be offered at least one of the two jobs is 0.727. d. What is the probability that Jennifer will be offered the first job but not the second job? This can be calculated by subtracting the probability of both jobs from the probability of being offered the first job only. Probability of being offered first job only = P(first job) - P(both jobs) = 0.380 - 0.2128 = 0.1672 Rounded to three decimal places, the probability that Jennifer will be offered the first job but not the second job is 0.167. e. What is the probability the Jennifer will not be offered the first job but will be offered the second job? This can be calculated by subtracting the probability of both jobs from the probability of being offered the second job only. Probability of being offered second job only = P(second job) - P(both jobs) = 0.560 0.2128 = 0.3472 Rounded to three decimal places, the probability that Jennifer will not be offered the first job but will be offered the second job is 0.347. Question 2 Alex and Sam purchase a foreclosed property for $50,000 and spend an additional $27,000 fixing up the property. They feel that they can sell the property for $120,000 with probability 0.15, $100,000 with probability 0.45, $80,000 with probability 0.25, and $60,000 with probability 0.15. a) What is the expected revenue Alex and Sam will make selling the property? b) What were their total costs in purchasing and improving the property? c) What is their total expected profit, which is the amount of revenue they expect to make minus the total costs of purchase and improvement? Answer Based on the information given, we can calculate the expected revenue, total costs, and expected profit for Alex and Sam's investment in the foreclosed property. a) To calculate the expected revenue, we need to multiply the possible selling prices with their corresponding probabilities and then add them together. So, the expected revenue is: Expected Revenue = ($120,000 * 0.15) + ($100,000 * 0.45) + ($80,000 * 0.25) + ($60,000 * 0.15) = $18,000 + $45,000 + $20,000 + $9,000 = $92,000 b) The total costs for purchasing and improving the property is $50,000 + $27,000 = $77,000. c) To find the total expected profit, we need to subtract the total costs from the expected revenue. So, the total expected profit is: $92,000 - $77,000 = $15,000. Question 3 For three events A, B, and C, establish that P((A ∪ B)|C) = P(A|C) + P(B|C) − P((A ∩ B)|C). Answer To establish the above equation, we first need to understand the concept of conditional probability. Conditional probability is the likelihood of an event occurring, given that another event has already occurred. Mathematically, it is represented as P(A|B), where A and B are events. Now, let's break down the given equation and understand it step by step. It is given that we have three events A, B, and C. Let's represent them on a Venn diagram. A ∪ B represents the union of events A and B, which is the combination of all possible outcomes from both events. In other words, it is the shaded area in the diagram where both events A and B intersect. Similarly, A ∩ B represents the intersection of events A and B, which is the common area between the two events. Thus, the equation P((A ∪ B)|C) = P(A|C) + P(B|C) − P((A ∩ B)|C) can be rewritten as P(A or B occurring, given C) = P(A occurring, given C) + P(B occurring, given C) − P(A and B both occurring, given C) This equation can be proven using the concept of conditional probability and the fact that the sum of probabilities of mutually exclusive events is equal to their union. First, let's consider the event A ∩ B. This event represents the intersection of events A and B, or the common area between the two events. Since A and B are mutually exclusive, the probability of A ∩ B is equal to 0. Therefore, P((A ∩ B)|C) = 0. Next, let's consider the event A or B. To find the probability of this event occurring given C, we can use the general multiplication rule which states that P(A ∪ B|C) = P(A|C) + P(B|C) − P(A and B|C). Since the probability of A and B both occurring, given C (represented as P(A and B|C)) is 0, we can eliminate it from the equation, leaving us with P(A ∪ B|C) = P(A|C) + P(B|C). This proves the given equation P((A ∪ B)|C) = P(A|C) + P(B|C) − P((A ∩ B)|C). Question 4 At least one of the process variables is controlled by the equipment POSSIBLE ANSWERS: Adaptive Control Mechanized Mechanized and Adaptive control Process control Semi-automatic Answer Based on the given question, it can be observed that the process in consideration involves the use of equipment that is responsible for controlling at least one process variable. In this context, the process variable can refer to any physical or chemical property of the system that can be measured and controlled to achieve the desired outcome. In order to choose the most appropriate answer, it is important to understand the meaning and significance of each of the possible options provided. 1. Adaptive Control: This refers to a type of control system that is capable of adjusting its behavior or parameters to adapt to changing conditions or to achieve a desired performance. In other words, it is a self-tuning control system that can learn and make adjustments based on feedback from the process being controlled. This type of control is often used in complex processes where the conditions are highly variable. 2. Mechanized: This refers to a process where machines or automation are used to perform tasks that were previously carried out manually. It typically involves the use of mechanical devices, such as motors, gears, and levers, to perform work. Mechanization can greatly increase the efficiency and productivity of a process. 3. Mechanized and Adaptive control: This refers to a combination of both mechanization and adaptive control. It involves the use of machines or automation in conjunction with a self-tuning control system to achieve the desired outcome. This type of control provides the benefits of both mechanization and adaptive control, leading to improved efficiency and adaptability. 4. Process control: This refers to the use of control systems to monitor and adjust various process variables in order to maintain a desired output. It involves the use of sensors, controllers, and actuators to measure and manipulate process variables, in order to achieve quality, efficiency, and safety goals. 5. Semi-automatic: This refers to a process that involves some level of human involvement, but also utilizes automated equipment to perform certain tasks. In semiautomatic processes, humans are responsible for controlling and monitoring the equipment, while the equipment performs certain functions on its own. Based on the above explanations, it can be concluded that adaptive control, mechanized and adaptive control, and process control are all suitable answers for the given question. Each of these options involves the use of equipment to control at least one process variable, with varying levels of automation and adaptability. However, depending on the specific parameters and requirements of the process in question, one of these options may be more appropriate than the others. Question 5 write a hypothetical case of two pages for the As a consultant in procurement audit and investigation, you have been contracted by Guy Guy company limited to train their staff to understand how fraudulent acts are concealed and how these acts can be unravelled Using the underlisted procurement fraud schemes Conflicts of interest ii. Phantom vendors iii. Split procurement orders/split orders iv. Kickbacks v. Personal purchases and • Explain how these schemes are concealed • Explain how these fraud schemes can be unravelled Answer Hypothetical Case: Uncovering and Preventing Procurement Fraud As a consultant in procurement audit and investigation, I have been contracted by Guy Guy Company Limited in order to train their staff on understanding how fraudulent acts are concealed and how these acts can be unravelled. In this hypothetical case, I will outline the various procurement fraud schemes that commonly occur and explain how they are concealed, as well as provide practical solutions on how these schemes can be unravelled. Case Background: Guy Guy Company Limited is a medium-sized retail chain that specializes in selling electronics. The company has been in business for over 20 years and has a reputation for providing high-quality products at competitive prices. Recently, the company has been experiencing a decline in profits and upon further investigation, it was revealed that there have been several procurement fraud incidents occurring within the company. As a result, the company's management has decided to take action by hiring a consultant to train their staff on how to detect and prevent fraudulent activities in the procurement process. Procurement Fraud Schemes: 1. Conflicts of Interest: A conflict of interest occurs when an employee misuses their position in a company for personal gain. This can happen when an employee has a financial interest in a company that is supplying goods or services to their employer. In this case, the employee may influence the decision-making process in order to benefit themselves or their family members. 2. Phantom Vendors: Phantom vendors are fictitious companies that are used to issue fake invoices and receive payments from the company. This scheme is often used in collusion with an employee, who will approve payments to these phantom vendors in return for a kickback. 3. Split Procurement Orders/Split Orders: Split procurement orders involve dividing a large purchase into several smaller purchases to avoid the need for management approval. In this way, the employee(s) involved can disguise their fraudulent activities and avoid detection. 4. Kickbacks: A kickback is a form of bribery in which an employee receives money or other benefits in return for influencing the procurement process in favor of a certain supplier. These kickbacks can be in the form of cash, gifts, or even lavish vacations. 5. Personal Purchases: Employees may use their position and access within the company to make personal purchases with the company's funds. This can involve using company credit cards for personal expenses or manipulating the procurement process to purchase personal items at a lower cost. Concealment of Procurement Fraud Schemes: In order to conceal these fraudulent activities, employees may employ various tactics such as creating falsified documentation, altering invoices and purchase orders, or manipulating the accounting system. In some cases, employees may also collude with external parties such as suppliers or vendors to create a false paper trail and cover up their actions. Unravelling Procurement Fraud Schemes: To effectively unravel these procurement fraud schemes, it is crucial for an organization to have strong internal controls in place. These controls can include strict policies and procedures for approving purchases, conducting thorough background checks on suppliers, and implementing a system of checks and balances. Employee training and awareness programs can also equip staff with the skills to detect and report any suspicious activities. In conclusion, it is essential for organizations to be aware of the various procurement fraud schemes and take proactive measures to prevent and detect them. By implementing strong internal controls and conducting regular audits, companies can mitigate the risk of falling victim to fraudulent activities and protect their financial interests. It is our responsibility as professionals in the procurement field to stay vigilant and continuously educate ourselves and our colleagues on the importance of ethical procurement practices.