PROBABILITY AND RANDOM VARIABLES

Group - I State the three axioms of probability with suitable examples. Define conditional probability. How is it different from simple probability? What is the significance of the Total Probability Theorem? Group - II Write Bayes’ theorem and mention any two real-life applications. Define a random variable. Why is it called “random”? Distinguish between discrete and continuous random variables. Group - III Define moments. What information does the second moment give? What is a Moment Generating Function (MGF)? In a class, 60% of students like Mathematics, 45% like Physics, and 30% like both. Group - IV Find the probability that a randomly chosen student likes at least one of the subjects. 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. 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. Group - V 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.