Question 1: Probability concepts and application [20] 1.1 .Consider the following random statement: \"I'm 110% sure that I will complete my MBA degree in 2022\" Using the basic laws of probability, explain why the latter statement is either true or false. (1) 1.2.What are the 3 way that are mainly used in statistical application to represent the probability distribution of a random variable? (1) 1.3.Consider the following: (8) a. 2 3 4 6 0.15 0.20 0.40 0.35 b. 0 l 2 3 0.2 0.3 0.3 0.2 c. __ 0.25 0.50 0 20 d. (i) Explain why each of the above (i.e. a. to d.) is or is not a valid probability distribution. (ii) Draw the histogram of the valid probability distribution(s) using MS Excel or any other software of your choice and comment on the skewness of the distribution (take a screenshot showing the formulas used no marks will be awarded if this is not shown). 1.4. According to a certain study on downloading of apps in a cell phone, majority of adult cell phone owners have downloaded an application ("app") to their cell phone. The following table gives the probability distribution for x, the number of apps used by cell phone owners who have downloaded an app to their cell phone per week. p(x) 0.30 0.24 0.06 0.06 0.05 y 0.07 0.05 0.03 0.02 10 0.02 (i) What is the value of y that will ensure that the properties of the probability distribution for a discrete random variable are satisfied? (1) (ii) Calculate the probability that the number of apps used by cell phone owners exceed 3. (2) (iii) Using MS Excel, calculate the expected value, variance and standard deviation of the number of apps used by cell phone owners. (3) 1.5. A lathe is a machining tool that is used primarily for shaping metal or wood, see the following figure (i.e. a Ryobi branded one): Ace Machine Works estimates that the probability its lathe tool is properly adjusted is 0.9. When the lathe is properly adjusted there is a 0.98 probability that the parts produced pass inspection. If the lathe is out of adjustment, the probability of a good part being produced is 0.1. A part randomly chosen is inspected and found to be acceptable. At this point, what is the posterior probability that the lathe tool is properly adjusted? (4)