A company making aircraft wings has a new automated process for producing the aileron (the part of the wing that helps to roll the aircraft). It is quicker than the previous manual process but is more prone to defects. Any imperfections in the metal can increase the chances of a stress fracture during flight. Thus, the company also wishes to introduce a new test for each aileron with an ultrasound machine to try to ensure that no unacceptable density deviations are present in ailerons fitted to aircraft. From trial data, the chances of an imperfection in an aileron manufacturing process are around 1%. The sensitivity (true positive rate) and specificity (true negative rate) of the new ultrasound test is 96% and 99% respectively. The company makes 200 ailerons per year. After using the test for 500 ailerons, the company obtain the following data plotted in Figure 1 below. If a potential defect is flagged by the ultrasound machine then a 1 is recorded in the database, and that piece of metal is removed from production and does not go on to produce an aileron. Flagged by ultrasound 0 50 100 150 200 250 300 350 400 450 500 Aileron number Figure 1: Flags raised by ultrasound machine after the manufacture of 500 ailerons. a) If no test was put in place to detect imperfections, calculate the probability per year of at least one defective aileron being fitted to an aircraft. [4 marks] b) Given the ultrasound machine is used to test for imperfections, draw a tree diagram showing the possible outcomes of the test. Calculate the probabilities of the possible outcomes. [6 marks] c) Recalculate the probability per year of at least one defective aileron being fitted to an aircraft, given all flagged pieces of metal are removed from production. [4 marks] d) What is the average number of flagged ailerons per year after the ultrasound test is introduced? In addition, write out an appropriate formula for the probability density function of the number of flagged ailerons per year. [5 marks] e) Using the data regarding flagged ailerons in Figure 1, how many of these flagged ailerons identified by the ultrasound machine would you expect to be true defects? [6 marks]