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Homework 1 1. In the design of an electromechanical product, seven different components are to be stacked into a cylindrical casing that holds 12 components in a manner that minimizes the impact of shocks. One end of the casing is designed as the bottom and the other end is the top. (a) How many different designs are possible? (b) If the seven components are all identical, how many different design are possible? (c) If the seven components consist of three of one type of component and four of another type, how many different designs are possible? 2. A computer system uses passwords that are six characters and each character is one of 26 letters (a‐z) or 10 integers (0‐9). Uppercase letter are not used. Let A denote the event that a password begins with a vowel (a,e,i,o,u) and let B denote the event that password ends with and event number (0,2,4,6,8). Suppose a hacker selects a password at random. Determine the following probabilities: (a) P(A) (b) P(B) (c) P(A∩ B) (d) P(A ∪B) 3. A batch h of 500 contaainers for frozen orange ju uice contains 5 that are deefective. Two are selected, at random, w without replaacement from m the batch. (aa) What is the e probability tthat the second one selectted is defectivve given that the first one was defective?? (b b) What is the e probability tthat both aree defective? (cc ) What is the e probability that both aree acceptable?? Three con ntainers are selected, at raandom, witho out replacemeent, from the batch (d d) What is the e probability tthat the third d one selected d is defective given that th he first and th he second on nes selected w were defectivve? (ee) What is the e probability tthat the third d one selected d is defective given that th he first one selected w was defective e and the seco ond one seleccted was okayy? (ff) What is the probability that all three are defectivee? 4. The pro obability that each device functions is aas shown. Asssume that thee probability tthat each devvice is functional does not de epend on wheether or not o other devices are functionaal. What is the probability that the circuitt operates? 5. Software to detect fraud in customer phone cards tracks the number of metropolitan areas where calls originate each day. It is found that 1% of the legitimate users originate calls from two or more metropolitan areas in a single day. However, 30% of fraudulent users originate calls from two or more metropolitan areas in a single day. The proportion of fraudulent users is 0.01%. If the same user originates calls from two or more metropolitan areas in a single day, what is the probability that the user is fraudulent? Homework 2 1. Heart failure is due to either natural occurrences (87%) or outside factor (13%). Outside factors are related to induced substances or foreign objects. Natural occurrences are caused by arterial blockage, disease, and infection. Suppose that 20 patients will visit an emergency room with heart failure. Assume that causes of heart failure between individuals are independent. (a) What is the probability that three individuals have conditions caused by outside factors? (b) What is the probability that three or more individuals have conditions caused by outside factors? (c) What is the mean and sd. Of the number of individuals with conditions caused by outside factors? 2. A trading company has eight computers that it uses to trade on the New York Stock Exchange (NYSE). The probability of a computer failing in a day is 0.005 and the computer fail independently. Computers are repaired in the evening and each day is an independent trial. (a) What is the probability that all eight computers fail in a day? (b) What is the mean number of a days until a specific computer fails? (c) What is the mean number of a days until all eight computer fails in the same day? 3.The number of flaws in bolts of cloth in textile manufacturing is assumed to be Poisson distributed with a mean of 0.1 flaw per square meter. (a) What is the probability that there are two flaws in 1 square meter of cloth? (b) What is the probability that there is one flaw in 10 square meters of cloth? (c) What is the probability that there is no flaws in 20 square meters of cloth? (d) What is the probability that there are at least 2 flaws in 10 square meters of cloth? 4. A congested computer network has a 1% chance of losing a data packet and packet losses are independent events. An email message requires 100 packets. (a) What is the distribution of data packets that must be resent? Include the parameter values. (b) What is the probability that two or more packets must be sent? (c) What is the mean and sd. of the number of packet that must be resent? (d) If there are 10 messages and each contains 100 packets, what is the probability that at least one message requires two or more packets be resent? Homework 3 1. The compressive strength of samples of cement can be modeled by a normal distribution with a mean of 6000 Kg/cm2 and sd. of 100 Kg/cm2 a) What is the probability that a sample’s strength is less than 6250 Kg/cm2? b) What is the probability that a sample’s strength is between 5800 and 5900 Kg/cm2? c) What strength is exceeded by 95% of the samples? 2. The manufacturing of semiconductor chips produces 2% of defective chips. Assume the chips are independent and that a lot contains 1000 chips. a) Approximate the probability that more than 25 chips are defective? b) Approximate the probability that between 20 and 30 chips are defective? 3. The time between arrivals of small aircraft at a county airport is exponentially distributed with a mean of one hour. a) What is the probability that more than three aircraft arrive within an hour? b) If 30 separate one‐hour intervals are chosen, what is the probability that no interval contains more than three arrivals? c) Determine the length of an interval of time(in hours) such that the probability that no arrivals occur during the interval is 0.10