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A university police department receives an average of \(3.7\) reports per week of lost student ID cards. a. Find the probability that at most 1 such report will be received during a given week by this police department. Use the Poisson probability distribution formula. 2\. Using the Poisson probabilities table, find the probability that during a given week the number of such reports received by this police department is i. 1 to 4 ii. at least 6 iii. at most 3

Based on its analysis of the future demand for its products, the financial department at Tipper Corporation has determined that there is a \(.17\) probability that the company will lose \(\$ 1.2\) million during the next year, a \(.21\) probability that it will lose \(\$ .7\) million, a \(.37\) probability that it will make a profit of \(\$ .9\) million, and a \(.25\) probability that it will make a profit of \(\$ 2.3\) million. a. Let \(x\) be a random variable that denotes the profit earned by this corporation during the next year. Write the probability distribution of \(x\). b. Find the mean and standard deviation of the probability distribution of part a. Give a brief interpretation of the value of the mean.

Spoke Weaving Corporation has eight weaving machines of the same kind and of the same age. The probability is .04 that any weaving machine will break down at any time. Find the probability that at any given time a. all eight weaving machines will be broken down b. exactly two weaving machines will be broken down c. none of the weaving machines will be broken down

Uniroyal Electronics Company buys certain parts for its refrigerators from Bob's Corporation. The parts are received in shipments of 400 boxes, each box containing 16 parts. The quality control department at Uniroyal Electronics first randomly selects 1 box from each shipment and then randomly selects 4 parts from that box. The shipment is accepted if at most 1 of the 4 parts is defective. The quality control inspector at Uniroyal Electronics selected a box from a recently received shipment of such parts. Unknown to the inspector, this box contains 3 defective parts. a. What is the probability that this shipment will be accepted? b. What is the probability that this shipment will not be accepted?

Suppose the owner of a salvage company is considering raising a sunken ship. If successful, the venture will yield a net profit of \(\$ 10\) million. Otherwise, the owner will lose \(\$ 4\) million. Let \(p\) denote the probability of success for this venture. Assume the owner is willing to take the risk to go ahead with this project provided the expected net profit is at least \(\$ 500,000\). a. If \(p=.40\), find the expected net profit. Will the owner be willing to take the risk with this probability of success? b. What is the smallest value of \(p\) for which the owner will take the risk to undertake this project?

Two teams, \(\mathrm{A}\) and \(\mathrm{B}\), will play a best-of-seven series, which will end as soon as one of the teams wins four games. Thus, the series may end in four, five, six, or seven games. Assume that each team has an equal chance of winning each game and that all games are independent of one another. Find the following probabilities. a. Team A wins the series in four games. b. Team A wins the series in five games. c. Seven games are required for a team to win the series

York Steel Corporation produces a special bearing that must meet rigid specifications. When the production process is running properly, \(10 \%\) of the bearings fail to meet the required specifications. Sometimes problems develop with the production process that cause the rejection rate to exceed \(10 \%\). To guard against this higher rejection rate, samples of 15 bearings are taken periodically and carefully inspected. If more than 2 bearings in a sample of 15 fail to meet the required specifications, production is suspended for necessary adjustments. a. If the true rate of rejection is \(10 \%\) (that is, the production process is working properly), what is the probability that the production will be suspended based on a sample of 15 bearings? b. What assumptions did you make in part a?

Many of you probably played the game "Rock, Paper, Scissors" as a child. Consider the following variation of that game. Instead of two players, suppose three players play this game, and let us call these players A, B, and C. Each player selects one of these three items - Rock, Paper, or Scissors - independent of each other. Player A will win the game if all three players select the same item, for example, rock. Player B will win the game if exactly two of the three players select the same item and the third player selects a different item. Player C will win the game if every player selects a different item. If Player \(\mathrm{B}\) wins the game, he or she will be paid \(\$ 1\). If Player \(C\) wins the game, he or she will be paid \(\$ 3\). Assuming that the expected winnings should be the same for each player to make this a fair game, how much should Player A be paid if he or she wins the game?

Customers arrive at the checkout counter of a supermarket at an average rate of 10 per hour. and these arrivals follow a Poisson distribution. Using each of the following two methods, find the probability that exactly 4 customers will arrive at this checkout counter during a 2-hour period. a. Use the arrivals in each of the two nonoverlapping 1 -hour periods and then add these. (Note that the numbers of arrivals in two nonoverlapping periods are independent of each other.) b. Use the arrivals in a single 2 -hour period.