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Probability, Statistics and Queueing

Learn probability, statistics and queueing with multiple-choice questions centered on calculations: permutations and combinations, the inclusion-exclusion principle, complementary events and conditional probability, Bayes' theorem, expected value, variance and standard deviation, the binomial, Poisson, uniform, normal and exponential distributions, the median, quartiles and box plots, the correlation coefficient and regression lines, the standard error and confidence intervals, hypothesis testing and p-values, queues (M/M/1) and Little's law, and the probabilities of bit errors and sampling inspection.

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Queueing (M/M/1) →

Probability, Statistics and Queueing: key points

"What is the probability that it happens at least once?" "What is the probability that a part judged positive is really defective?" "How much longer does the wait get when server utilization rises from 80% to 90%?" Probability and statistics answer questions like these. They are the common tools for handling variation and uncertainty with numbers in many fields of engineering: semiconductor yield and reliability, bit errors in communication, computer performance evaluation, machine learning and more.

This topic covers counting cases with permutations and combinations, the inclusion-exclusion principle, complementary events and independence, conditional probability and Bayes' theorem, expected value and its linearity, and variance and standard deviation. It then checks the properties of the binomial, Poisson, uniform, normal (the 68-95-99.7 rule and standardization) and exponential (memorylessness) distributions through calculations. On the statistics side, it deals with the median, quartiles and box plots, the correlation coefficient and the slope of the regression line, the standard error and confidence intervals, and p-values and type I and type II errors in hypothesis testing. The last part calculates utilization, average waiting time and response time with the M/M/1 queueing model, covers Little's law and the effect of how queues are formed, and applies these ideas to the probabilities of bit errors, parity checks and sampling inspection. It helps with the applied mathematics (probability, statistics and queueing) of IT certification exams such as Japan's Fundamental and Applied Information Technology Engineer Examinations. Control charts and process capability indices are covered in "Yield and quality control", failure rates and the Weibull distribution in "Reliability", confusion matrices and machine learning models in "Machine Learning Basics", and availability calculations in "System Architecture and Reliability".

All Probability, Statistics and Queueing questions (40)
  1. Three committee members are chosen from 8 people. If the members have no distinct roles, which of the following is the number of possible choices?
  2. A chair, a vice-chair and a secretary are chosen, one each, from 5 people. If no one may hold two or more roles, which of the following is the number of possible choices?
  3. A 4-digit PIN is made by arranging four of the digits 0 to 9. If the same digit may be used any number of times and the first digit may be 0, which of the following is the number of possible PINs?
  4. In a town whose streets form a square grid, you go from intersection A to intersection B, which is 4 blocks east and 3 blocks north of A, by a shortest route that moves only east or north. Which of the following is the number of routes?
  5. Of 200 chips inspected, 30 had defect X and 20 had defect Y, and 8 of these had both X and Y. Which of the following is the number of chips with neither defect?
  6. When a die is rolled 3 times, which of the following is closest to the probability that a 6 comes up at least once?
  7. Among 10 lottery tickets, 3 are winners. Two tickets are drawn one after the other, and a drawn ticket is not put back. Which of the following is the probability that both are winners?
  8. A fab processed 500 wafers in a month, 100 of them in chamber 2. A total of 40 wafers failed, and 15 of these had been processed in chamber 2. Which of the following is the conditional probability that a wafer processed in chamber 2 fails?
  9. In a population of chips, 1% have a latent defect. A screening test judges 95% of the defective chips positive and wrongly judges 4% of the good chips positive. Which of the following is closest to the probability that a chip judged positive is actually defective?
  10. There are 1,000 lottery tickets at ¥50 each: one wins a ¥10,000 prize, ten win ¥1,000 prizes, and the rest win nothing. Which of the following is the expected value of the prize per ticket?
  11. A chip costs ¥300 to make. A good chip sells for ¥500, and a defective chip cannot be sold and is discarded. If the probability that a chip is good is 80%, which of the following is the expected profit per chip made?
  12. Ten keys are stored in a hash table with 100 buckets. Each key is assigned independently to one of the 100 buckets with equal probability. Which of the following is the expected number of pairs of keys that are assigned to the same bucket?
  13. Taking the variance as the sum of the squared differences from the mean divided by the number of values, which of the following is the standard deviation of the 8 data values 2, 4, 4, 4, 5, 5, 7, 9?
  14. Five products are taken from a process with a defect rate of 20%. Whether each product is defective is independent of the others. Which of the following is closest to the probability that exactly one of the five is defective?
  15. Inquiries to a service desk arrive at random, independently of one another, at an average of 3 per hour, and the number arriving in one hour follows a Poisson distribution. Taking e⁻³ ≈ 0.0498, which of the following is closest to the probability that exactly 2 arrive in a given hour?
  16. Requests reach a server as Poisson arrivals (independent, random arrivals) at an average of 6 per minute. Which of the following correctly combines the distribution followed by the intervals between arrivals and their mean?
  17. The time to failure of a piece of equipment follows an exponential distribution with a mean of 1,000 hours. Take e^(−0.5) ≈ 0.61, e^(−1) ≈ 0.37 and e^(−1.5) ≈ 0.22. Given that the equipment has run for 500 hours without failure, which of the following is closest to the probability that it runs at least another 1,000 hours without failure?
  18. You arrive at a bus stop served every 10 minutes at a random time, without checking the timetable. The waiting time follows a uniform distribution from 0 to 10 minutes. Which of the following correctly combines the expected waiting time and the probability that the wait is 3 minutes or less?
  19. The dimension of a part follows a normal distribution with a mean of 50.0 mm and a standard deviation of 0.1 mm. Which of the following is closest to the fraction of parts whose dimension lies between 49.7 mm and 50.3 mm?
  20. A scored 80 on test X, and B scored 75 on test Y. Test X had a mean of 65 and a standard deviation of 10, and test Y had a mean of 60 and a standard deviation of 6. Which of the following correctly compares their relative positions within their own tests using the standardized value z = (score − mean) ÷ standard deviation?
  21. The delay of a circuit path varies from chip to chip following a normal distribution with a mean of 940 ps and a standard deviation of 30 ps. Taking the fraction of a normal distribution within the mean ±2σ to be about 95.4%, which of the following is closest to the fraction of chips whose delay exceeds the clock period of 1,000 ps?
  22. The response time of a server was measured 5 times, giving 20, 3, 5, 3 and 4 milliseconds in the order measured. Which of the following is the median of these data?
  23. The 11 data values 1, 3, 4, 6, 7, 8, 10, 12, 13, 15, 20 are shown in a box plot. The first and third quartiles are taken as the medians of the lower and upper halves of the data, excluding the median. Which of the following is the interquartile range, which corresponds to the length of the box?
  24. Which of the following correctly describes the correlation coefficient of two quantities?
  25. Which of the following is closest to the correlation coefficient of the five data pairs (x, y) = (1, 2), (2, 4), (3, 5), (4, 4), (5, 5)?
  26. Two quantities x and y have a correlation coefficient of 0.8, the standard deviation of x is 5, and the standard deviation of y is 10. Which of the following is the slope of the least-squares regression line predicting y from x?
  27. A sample of size 36 is drawn at random from a population with a standard deviation of 12. Which of the following is the standard error of the sample mean (the standard deviation that describes the spread of the sample mean)?
  28. Which of the following correctly describes the meaning of a 95% confidence interval for the population mean obtained from a sample?
  29. The standard deviation of a population is known to be 10. A sample of size 100 drawn from it has a sample mean of 50.0. Taking the two-sided 95% value of the standard normal distribution to be 1.96, which of the following is the 95% confidence interval for the population mean?
  30. Which of the following correctly describes the meaning of the p-value obtained in a hypothesis test?
  31. Whether a product lot meets its specification is tested using a sample taken from it. If the null hypothesis is "this lot meets the specification", which of the following is a type I error?
  32. Requests arrive at a service window at an average of 12 per hour, and each takes an average of 4 minutes to handle. Treating it as an M/M/1 queue, which of the following is the utilization of the window?
  33. Requests arrive at a server at an average of 3 per minute, and each takes an average of 15 seconds to process. Treating the server as an M/M/1 queue, which of the following is the average time from the arrival of a request until its processing is finished (the average response time)?
  34. A web server holds an average of 12 requests, counting those being processed and those waiting. Requests arrive at 40 per second and are processed at the same rate (steady state). Which of the following is the average time a request spends in the server, found from Little's law?
  35. For a server that can be treated as an M/M/1 queue, the arrival rate increased with the average service time unchanged, and the utilization rose from 0.8 to 0.9. By what factor did the average waiting time Wq (the time spent waiting in the queue) change? Which of the following is closest?
  36. There are two service windows of the same speed. In scheme A, each customer picks one of the windows with equal probability on arrival and joins that window's own queue. In scheme B, customers join a single queue and are served by whichever window becomes free first. Arrivals are random, and the total arrival rate and service speed are the same in both schemes. Which of the following correctly compares the average waiting time?
  37. A server with an average service time of 0.2 seconds is treated as an M/M/1 queue. To keep the average response time (waiting time plus processing time) at or below 1.0 second, which of the following is closest to the maximum arrival rate that can be accepted?
  38. In an acceptance sampling inspection, 10 items are drawn at random from a large lot with a defect rate of 5%, and the lot is accepted if none of them is defective. Which of the following is closest to the probability that this lot is accepted?
  39. 10,000 bits of data are sent over a link with a bit error rate of 1 × 10⁻⁴. If the errors in each bit occur independently, which of the following is closest to the probability that at least one bit is in error?
  40. Eight data bits plus one even parity bit, 9 bits in total, are sent. If each bit is in error independently with a probability of 1%, which of the following is closest to the probability of an error that the parity check cannot detect?

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