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Chapter 5: Problem 40
What are the parameters of the binomial probability distribution, and what do they mean?
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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?
A household receives an average of \(1.7\) pieces of junk mail per day. Find the probability that this household will receive exactly 3 pieces of junk mail on a certain day. Use the Poisson probability distribution formula.
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.
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?
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?
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