Probability, 150 150 Affordable Capstone Projects Written from Scratch

Sample Midterm Exam

Calculation Part

  1. Please evaluate the following business case. One e-commerce company has recently learned that the cost of website downtime is approximately $50,000 per hour. Currently, the server system has about 95% of reliability.
  • Considering that the server systems usually run 24 hours a day except for planned maintenance, how many downtimes (in hours) is expected in a week? And what are the associated cost for service disruption?
  • To reduce the downtime, one consulting firm has suggested the firm use duplicated system design. In such design, an additional server that has the same reliability performance will be added and running at the same time. What is the probability that at least one server is running and how does it reduce the disruption cost?
  • How such probability changes if the company adopts a trio design (that is use three same servers)?
  • What information is needed to determine which design (single, duo, and trio) has the best cost effect?


  1. Please translate Chevalier’s gambling problem into binomial probability distribution language and then use the knowledge of distribution to calculate. Then, compare with the answer. Two bets are: (1) at least one six in 4 rolls and (2) at least one double-six in 24 rolls.


  1. Please consider following two variables.
  • Variable 1: 23, 25, 35, 28, 26, 29
  • Variable 2: 2440, 2690, 3450, 2690, 2780, 3110
  • Please draw scatter plot for the variables, and make comments about their relationship. Do not use covariance or correlation coefficients.
  • Does the order within the variable matters when it comes to evaluate the correlation between two? (that is, if I change the order of numbers for variable 1 and 2, will it still display the same correlation?) Why yes and why no?
  • If variable 1 is age, variable 2 is monthly salary, why comments can you make about two variables?
  • Can you construct a histogram for variable 2?
  • Now variable 2 is added for following values: 23, 340, 38493, 87695, 98043 (total 11 observations), how would you group them?


  1. I suspect 70% bad parts are from A supplier and 30% of bad parts are from B supplier. Historically, defective parts for supplier A was 3% and that for supplier B was 5%.
  • Please translate the problem into the language of probability (that is, notation).
  • What is the revised probability for the supplier A and B. (Or, put it in another way, if I found a bad part, what is the chance that it is from A and B, respectively.)
  • How did it change my initial assessment?
  1. Please answer following questions regarding distributions.
  • What information does z value tell us? What is the role of σin the process?
  • If a dataset has σof zero, what is the range of the dataset?
  • A certain manufacturing process requires the milling process be aimed at μ= 12 and σ= 3. If the actual process has the μ= 12 and σ= 4. What will happen? What about μ = 11 and σ = 2? Please draw and answer. (hint: this question may not need calculations)
  • Use μ= 12 and σ = 3, please find out following probabilities (draw and show the process).
    1. x≥15
    2. x≤9
    3. 12≤x≤9