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- 2. Iron bars produced by a machine can either be classified as "Good" with probability p, or "Bad" with probability q = 1- p. The quality of each iron bar produced is independent of each other. (a) Suppose there are n iron bars produced by the machine. Write down the probability that there are exactly r "Good" iron bars among the n of them for all positive integers r. (b) Suppose the machine keeps producing iron bars until k "Good" ones are obtained. Let X be the total number of iron bars produced. Show that the probability mass function of X is given by Px(x) = (x − 1) ₁² (1 - p)² -*. x=k, k +1,.... (c) In a single day, the machine produces exactly N = n iron bars, where N has mean and variance given by E(N) = a, Var(N) = 3. Let W be the total number of "Good" iron bars produced in a single day. i. Find E(W) and Var(W) in terms of p, a and 3. You can use the mean and variance of any standard distributions without proof as long as you state them clearly. ii. Determine the covariance…9 A bag contains 12 red and 8 blue balls that are identical except for their colour. A ball is chosen from the bag, its colour noted, and then it is replaced. Eight balls are chosen. (i) Find (a) the probability of obtaining five red balls (b) the probability of obtaining at least two blue balls (c) the probability of obtaining equal numbers of red and blue balls. (ii) How many balls need to be chosen if the probability of obtaining at least one red ball is to be at least 99%?0. A box has 10 coins of which, 6 coins are fair, 4 coins are loaded such that the probability of getting a tail is 1/3 with each loaded coin. A coin is selected from the box at random and is tossed once with the result of tail. Find the conditional probability that the coin is fair.
- Please help me with this discrete II problem. Thank you1.) A basket contains 11 red apples and 8 green apples. A sample of 5 apples is drawn. Find the probability that the sample contains the following, and give your answer as a reduced fraction. (a) All green apples (b) 2 green apples and 3 red apples (c) More red than green apples6. Someone plans to travel at the weekend. According to the weather forecast, the probability of raining on the first and the second day of weekends is 0.5 and 0.3 respectively, the probability of raining on both days is 0.2. Try to find: (Let Ai be event of raining on the ith day, i = 1,2, then P (A₁) = 0.5, P (A2) = 0.3, P (A1 A2) = 0.2.) (1) The probability of raining on the first day but not on the second day; (2) The probability of raining on one of the two days; (3) The probability of raining at least on one day.
- 5. a) Suppose you have three coins in a bag: the first coin is fair, with prob- ability of HEAD (H) equals to that of TAIL (T), i.e., P(X = H|M1) = P(X = T|M1), where X is a binary random variable representing HEAD or TAIL. The second is a loaded coin with P(X = takes a coin out of the bag at random and flips it. When it lands, you observe that the side of the coin facing up is HEAD. What is the probabilities that this is the first, second and third coin? Hint: You may assume the three coins are equal likely to be selected from the bag, i.e., P(M1) = P(M2) = P(M3) = , and you need to calculate P(M||X = H), P(M2|X = H) and P(M3|X = H). H|M2) = 0.8. The third is also a loaded coin with P(X = Н Мз) — 0.2. Your friend 6) What if we don't know the probabilities of HEAD for each coin? Design an iterative algorithm using pseudo-code to estimate these probabilities.2. A student majoring in phycology is trying to decide on the number of firms to which he should apply. Given his work experience and grades he can expect to receive a job offer from 75% of the firms he applies. The student decides to apply to only 4 firms. What is the probability that he receives?a) No job offers P(x = 0) b) Less than 2 job offers P(x <2) c) At least 2 job offers P(x ≥ 2)What is the likelihood of selecting a 4-digit number in the win-4 lottery, where one digit appears exactly three times, given that each digit from 0 to 9 has an equal probability of being drawn in each of the four draws? Note that the number starts with 0 is a valid 4 digit-number such as 0001 is an option