Week 4 (29-12-25 to 02-01-26)

Prepare

📖 Read the syllabus

📖 Read the support resources

Participate

🖥️ Lecture 3 -Continuous Probability Distributionss

🖥️ Statistical Table - Statistical Table for probability estimations.

🖥️ Lecture 2 - Discrete Probability Distributions

🖥️ Lecture 1 - Introduction to Probability

Practice

  1. In celebration of the Computer Engineering Department’s launch of a new high-performance computing lab, a feedback survey was conducted among students who attended the inauguration event. The survey revealed a strong 75% satisfaction rate, highlighting the positive reception of the new lab facilities and demonstrations. To better understand feedback trends, the department analyzed the probabilities associated with satisfaction levels in randomly selected groups of 15 student attendees.

    1. What is the probability of having exactly eleven satisfied attendees?

    2. Find the probability of encountering more than five satisfied attendees.

    3. Find the probability that there are twelve or fewer satisfied attendees.

  2. An automobile manufacturer is concerned about a fault in the braking mechanism of a particular model. The fault can, on rare occasions, cause a catastrophe at high speed. The distribution of the number of cars per year that will experience the catastrophe is a Poisson random variable with λ = 5

    1. What is the probability that at most 3 cars per year will experience a catastrophe?
    2. What is the probability that more than 1 car per year will experience a catastrophe?

  3. The Gold Credit card company knows that 70% of its customers are males. The company plans to promote its credit card by selecting a random sample of 5 customers each month and offering them a free vacation.

    1. What is the probability that in a given month all five selected customers are males?
    2. What is the probability that the selected sample includes at least one female?
    3. What is the mean number of male customers who are likely to be selected in the coming month?

  4. In the mechanical engineering laboratory, there are 20 hydraulic pressure gauges used for testing the performance of fluid power systems. A recent inspection reveals that 7 of these gauges are malfunctioning, which could lead to inaccurate pressure readings during critical experiments. Suppose 6 gauges are randomly selected for calibration testing:

    1. What is the probability that precisely three of them will be malfunctioning?
    2. What is the probability that none of the selected gauges are malfunctioning?

  5. A large company has an inspection system for the batches of small compressors purchased from vendors. A batch typically contains 15 compressors. In the inspection system, a random sample of 5 is selected and all are tested. Suppose there are 3 faulty compressors in the batch of 15.

    1. What is the probability that for a given sample there will be 1 faulty compressor?
    2. What is the probability that the inspection will reveal more than one faulty compressor in the sample?

  6. In the Department of Civil Engineering at the University of Jaffna, a newly established testing facility is used to assess the structural integrity of small concrete beam specimens. Due to limited resources, only a fixed number of specimens can be tested each day. The number of beam specimens waiting for testing on any given day has the following probability distribution:

    x 1 2 3 4 5 6
    P(X=x) 0.12 m 0.28 0.12 0.16 0.05

    Where X denotes the number of concrete beam specimens waiting for testing on a given day.

    1. Determine the value of m?
    2. What is the expected number of specimens waiting for testing?
    3. What is the standard deviation of the number of specimens?
    4. What is the probability that fewer than three specimens are waiting for testing?
    5. What is the probability that at least four specimens are waiting for testing?

Exercise

  1. It is conjectured that an impurity exists in 30% of all drinking wells in a certain rural community. In order to gain some insight into the true extent of the problem, it is determined that some testing is necessary. It is too expensive to test all of the wells in the area, so 10 are randomly selected for testing.

    1. Using the binomial distribution, what is the probability that exactly 3 wells have the impurity, assuming that the conjecture is correct?
    2. What is the probability that more than 3 wells are impure?
    3. What is the mean and variance of the distribution of the number of drinking wells?

  2. An automatic welding machine is being considered for use in a production process. It will be considered for purchase if it is successful on 99% of its welds. Otherwise, it will not be considered efficient. A test is to be conducted with a prototype that is to perform 100 welds. The machine will be accepted for manufacture if it misses no more than 3 welds.

    1. What is the probability that a good machine will be rejected?
    2. What is the probability that an inefficient machine with 95% welding success will be accepted?

  3. An automobile manufacturer is concerned about a fault in the braking mechanism of a particular model. The fault can, on rare occasions, cause a catastrophe at high speed. The distribution of the number of cars per year that will experience the catastrophe is a Poisson random variable with λ = 5.

    1. What is the probability that at most 3 cars per year will experience a catastrophe?
    2. What is the probability that more than 1 car per year will experience a catastrophe?

  4. Changes in airport procedures require considerable planning. Arrival rates of aircraft are important factors that must be taken into account. Suppose small aircraft arrive at a certain airport, according to a Poisson process, at the rate of 6 per hour. Thus, the Poisson parameter for arrivals over a period of hours is µ = 6t.

    1. What is the probability that exactly 4 small aircraft arrive during a 1-hour period?
    2. What is the probability that at least 4 arrive during a 2-hour period?
    3. If we define a working day as 12 hours, what is the probability that at least 75 small aircraft arrive during a working day?

  5. A service engineer is can be called out for maintenance on the photocopiers in the offices of four large companies, A, B, C and D. On any given week there is a probability of 0.1 that he will be called to each of these companies. The event of being called to one company is independent of whether or not he is called to any of the others.

    1. Find the probability that, on a particular day,

      i. He is called to all four companies,

      ii. He is called to at least three companies,

      iii. He is called to all four given that he is called to at least one,

      iv. He is called to all four given that he is called to Company A.

    2. Find the expected value and variance of the number of these companies which call the engineer on a given day.

Perform

Assignment 1 Examination scheduled from 11.00 am to 11.40 am on January 2, 2026 (Friday). To ensure that the exam runs smoothly, we have arranged the exam halls as follows: 

  • Group 1 (Registration numbers 2024/E/001 to 2024/E/101) will be in Exam Hall 1 (First floor, Computer Engineering Department)  

  • Group 2 (Registration numbers 2024/E/102 to 2024/E/198and re-attempt students) will be in Exam Hall 2 (Drawing Hall 2nd floor, Admin Building).

Make sure to review all sections thoroughly to ensure you’re prepared for the exam! I hope everyone follows the exam policies and cooperates with the exam administration to make the exam run smoothly. Moreover, you can access class materials by checking out the course webpage.

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