Explorations Table 210 d 14 15 16 17 18 AOQ 006476 006535 006561 006556 006523

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Sampling here has had a dramatic impact on the average percentage of D items delivered to the customer

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This is just one example of how probability and statistics can assist in delivering high-quality product to consumers There are many other techniques used that are called in general statistical process control methods, or SPC; these have found wide use in industry today Statistical process control is the subject of 11

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We have explored permutations and combinations in this chapter and have applied them to several problems, most notably a plan for choosing the best candidate from a group of applicants for a position and acceptance sampling where we found that sampling does improve the quality of the product sold and actually puts a limit on the percentage of unacceptable product sold We will continue our discussion or production methods and the role probability and statistics can play in producing more acceptable product in the chapter on quality control and statistical process control We continue in the following two chapters with a discussion of conditional probability and geometric probability Each of these topics ts well into a course in geometry

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EXPLORATIONS

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1 Use the principle of inclusion and exclusion to prove the general addition theorem for two events 2 Find the probability a poker hand has (a) exactly two aces; (b) exactly one pair 3 What is the probability a bridge hand (13 cards from a deck of 52 cards) does not contain a heart 4 Simulate Example 28: Choose random permutations of the integers from 1 to 10 with 10 being the most quali ed assistant Compare the probabilities

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2

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of making the best choice by letting 1, 2, or 3 applicants go by and then choosing the best applicant better than the best of those passed by 5 Simulate Example 24: Choose random permutations of the integers from 1 to 5 and count the number of instances of an integer being in its own place Then count the number of derangements, that is, permutations where no integer occupies its own place 6 In Example 27, how would you use the maximum of the sample in order to estimate the maximum of the population 7 A famous correspondence between the Chevalier de Mere and Blaise Pascal, each of whom made important contributions to the theory of probability, contained the question Which is more likely at least one 6 in 4 rolls of a fair die or at least one sum of 12 in 24 rolls of a pair of dice Show that the two events are almost equally likely Which is more likely

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Conditional Probability

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to consider some problems involving conditional probability to show diagrams of conditional probability problems to show how these probability problems can be solved using only the area of a rectangle to show connections with geometry to show how a test for cancer and other medical tests can be misinterpreted to analyze the famous Let s Make a Deal problem

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A physician tells a patient that a test for cancer has given a positive response (indicating the possible presence of the disease) In this particular test, the physician knows that the test is 95% accurate both for patients who have cancer and for patients who do not have cancer The test appears at rst glance to be quite accurate How is it then, based on this test alone, that this patient almost certainly does not have cancer We will explain this I have a class of 65 students I regard one of my students as an expert since she never makes an error I regret to report that the remaining students are terribly ignorant of my subject, and so guess at each answer I gave a test with six true false questions; a paper I have selected at random has each of these questions answered correctly What is the probability that I have selected the expert s paper The answer may be surprising Each of these problems can be solved using what is usually known in introductory courses as Bayes theorem We will not need this theorem at all to solve these problems No formulas are necessary, except that for the area of a rectangle, so these problems can actually be explained to many students of mathematics

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