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[3] A chip manufacturing company manufactures chips independently. Each manufactured chip is either good or defective. Based on the company's data, with probability 0.18, one out of every two manufactured chips is defective. It is also known that the probability of a defective chip is smaller than 0.5. Find the probability that (a) [9 pts] more than two chips are defective out of 100 manufactured chips (b) [8 pts] the 34" chip and one other chip is defective out of the first 100 manufactured chips (c) [8 pts] 2 chips among the first 50 manufactured chips and I chip among the second 50 manufactured chips are defective6. Suppose that the probability of being tutored in Statistics is 0.55, while the probability of being tutored in Physics is 0.32, and the probability of being tutored in both Statistics and Physics is 0.09. a) What is the probability of being tutored in Statistics or Physics? b) What is the probability of not being tutored in Statistics nor Physics? c) Are the events "being tutored in Statistics" and "being tutored in Physics" mutually exclusive? Explain.[a) In a Galton-Watson branching process starting with one individual in generation zero, the offspring random variable X has a geometric distribution with mean 3: X ~ Go[p]. [ Find the value of the parameter p of the geometric distribution. [in) Calculate to four decimal places the probability that the process becomes extinct: By the third generation; [2) At the fourth generation. (iii) Calculate to four decimal places the probability that the process eventually becomes extinct. (b] Consider the branching process for which an individual has zero offspring with probability a, or 3 offspring with probability 1- a. The offspring random variable Y has the generating function n(s) = a - [1 - 0] $43. [i) For what values of a is extinction not certain? [ii) When a = 36, find the probability of extinction. (iin If there were four individuals in generation zero, what would be the probability that the process eventually becomes extinct?Problem 4 Each year, ratings are compiled concerning the performance of new cars during the first 90 days of use. Suppose that the cars have been categorized according to whether the car needs warranty-related repair (yes or no) and the country in which the company manufacturing the car is based (United States or not United States). Based on the data collected, the probability that the new car needs a warranty repair is 0.04, the probability that it was manufactured by a U.S-based company is 0.60, and the probability that the new car needs a warranty repair and was manufactured by a U.S. - based company is 0.025. a. Suppose you know that a company based in the United States manufactures a particular car. What is the probability that the car needs warranty repair
