PROBABILITY AND RANDOM VARIABLES

PART A: 1.State the three axioms of probability with suitable examples. 2.Define conditional probability. How is it different from simple probability? 3.What is the significance of the Total Probability Theorem? 4.Write Bayes’ theorem and mention any two real-life applications. 5.Define a random variable. Why is it called “random”? 6.Distinguish between discrete and continuous random variables. 7.Define moments. What information does the second moment give? 8.What is a Moment Generating Function (MGF)? PART B: 1.In a class, 60% of students like Mathematics, 45% like Physics, and 30% like both. Find the probability that a randomly chosen student likes at least one of the subjects. 2.Two machines A and B produce 40% and 60% of total items respectively. The percentage of defective items produced by A is 3% and by B is 5%. If an item selected at random is defective, find the probability that it was produced by machine B. 3.The probability that a student passes Mathematics is 0.7 and the probability that the student passes Physics given that he passes Mathematics is 0.8. Find the probability that the student passes both subjects. Case Study: A smart sensor detects faults in a system. It correctly detects a fault 95% of the time, but it also falsely indicates a fault 5% of the time when there is none. Only 2% of systems actually have faults. Tasks: Identify the events clearly. Use Bayes’ theorem to find the probability that a system actually has a fault when the sensor indicates a fault. Comment on the reliability of the sensor.