Week 2 (15-12-25 to 19-12-25)

Prepare

📖 Read the syllabus

📖 Read the support resources

Participate

🖥️ Lecture 2 - Discrete Probability Distributions

🖥️ Statistical Table - Statistical Table for probability estimations.

🖥️ Lecture 1 - Introduction to Probability

Practice

📋 Practice Problems

1. At a certain assembly plant, three machines make 30%, 45% and 25% respectively of the products. It is known from past experience that 2%, 3% and 4% of the products made by each machine, respectively, are defective. Now, suppose that a finished product is randomly selected. What is the probability that it is defective? And if a product was chosen randomly and found to be defective, what is the probability that it was made by machine 3?

2. Mr. Ragul, a network administrator at the University of Jaffna, is responsible for maintaining the university’s server infrastructure. He has three options for deploying a critical software update to the servers: using a local deployment tool, a cloud-based deployment service, or a manual deployment process. Due to varying network conditions and system loads, the probability of the deployment being delayed depends on the method chosen. Specifically: If he uses the local deployment tool, there is a 50% chance the deployment will be delayed due to high server load. If he uses the cloud-based deployment service, which has dedicated bandwidth, there is a 20% chance the deployment will be delayed. If he uses the manual deployment process, the probability of delay is only 5%. One day, the deployment is delayed. The IT manager wants to estimate the probability that Ragul used the local deployment tool that day.

   1. Suppose the IT manager assumes that Ragul is equally likely to use any of the three deployment methods (i.e., a 1/3 probability for each). Using Bayes’ theorem, what estimate does the IT manager obtain for the probability that Ragul used the local deployment tool?
   2. Now, suppose the IT manager knows that Ragul’s deployment habits are as follows: He uses the local deployment tool 20% of the time, the cloud-based deployment service 15% of the time, and the manual deployment process 65% of the time. Using Bayes’ theorem with this updated information, what estimate does the IT manager obtain for the probability that Ragul used the local deployment tool?

3. In a collaborative project at the Faculty of Engineering, University of Jaffna, three electrical engineering teams - Team Ohm, Team Faraday, and Team Tesla - contribute respectively 12%, 48%, and 40% of the circuit modules for a complex embedded system. Team Ohm, known for its high precision, produces faulty modules only 0.5% of the time. However, Team Faraday and Team Tesla have higher fault rates, with 5% and 8% of their modules being defective, respectively. All modules are integrated into a central system. During a quality assurance test, one module is randomly selected and found to be defective.

   1. What is the conditional probability that it originated from Team Ohm?
   2. What is the probability that it came from Team Faraday?

4. As part of their coursework on Reliability Engineering in the Department of Electrical and Computer Engineering at the University of Jaffna, students are analyzing a faulttolerant computer system in which three components K1, K2, and K3 are arranged in a sequential redundancy configuration. The system initially operates with component K1; if K1 fails, it switches to K2, and if K2 also fails, it finally switches to K3. The failures of these components are mutually independent, with failure probabilities of 0.01, 0.03, and 0.08 respectively. The system is considered functional as long as at least one of the three components works when needed. What is the probability that the system does not fail?

5. A group of 10 people consists of 3 managers and 7 employees. A team of 4 people to be selected randomly. What is the probability that:

   1. The team includes exactly 1 manager.
   2. The team has at least 1 manager.
   3. The team has 2 managers and 2 employee.

Exercises

1. A truth serum has the property that 90% of the guilty suspects are properly judged while, of course, 10% of the guilty suspects are improperly found innocent. On the other hand, innocent suspects are misjudged 1% of the time. If the suspect was selected from a group of suspects of which only 5% have ever committed a crime, and the serum indicates that he is guilty, what is the probability that he is innocent?

2. A construction company employs two sales engineers. Engineer 1 does the work of estimating cost for 70% of jobs bid by the company. Engineer 2 does the work for 30% of jobs bid by the company. It is known that the error rate for engineer 1 is such that 0.02 is the probability of an error when he does the work, where as the probability of an error in the work of engineer 2 is 0.04. Suppose a bid arrives and a serious error occurs in estimating cost. Which engineer would you guess did the work? Explain and show all work.

3. A group of 10 people consists of 3 managers and 7 employees. A team of 4 people to be selected randomly. What is the probability that

   1. the team includes exactly 1 manager.
   2. the team has at least 1 manager.
   3. the team has 3 managers and 1 employee.

4. A game involves drawing cards from three decks. Deck A has 3 blue cards and 2 green cards. Deck B contains 4 blue and 1 green cards. Deck C comprises 5 blue cards. A card is drawn from Deck A and placed in Deck C. A card from Deck B is drawn and placed in Deck C. Finally, a card is drawn from Deck C.

   1. Draw a tree diagram to illustrate this game with all probabilities
   2. Calculate the probability that exactly two green cards are drawn.
   3. Given that two green cards are drawn, find the probability that the card from Deck A is blue.

5. In an organization of 300 staff, they are divided among departments like this:

   	Design	Production	Quality Control
   Male	60	80	40
   Female	50	40	30

   1. What is the probability of selecting a Production employee given that a male was selected?

   2. What is the probability of selecting a male given that a Design employee was selected?

   3. If we randomly select one employee, let X be the event that the selected employee is a female and Y be the event that the selected employee is from the Quality control. Are events X and Y mutually exclusive?

6. 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?

7. 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?

8. 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?

9. 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?

10. 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. ```

```

Back to course schedule ⏎