2000%20spring%20solutions - Math 218 Final Exam Solved...

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Math 218 Final Exam Solved Spring 2000 The solutions which we reproduce below are far more elaborate than what was expected on the final. After all, not many students have laser printers and Mathematica running on a laptop during the exam! Our purpose is to give as full an explanation as possible, so you understand the solution. Problem 1 . (20 points) In a company of 200 employees, there are 32 employ- ees making at least $100,000 a year. There are 47 employees in the company that have a graduate degree. There are 143 employees that do not have a graduate degree and earn less than $100,000 per year. (a) Find the probability that a randomly chosen employee makes less than $100,000 and has a graduate degree. (b) Find the probability that a randomly chosen employee makes at least $100,000 or has a graduate degree. (c) If an employee is selected at random, let A be the event that the em- ployee makes less than $100,000, and let B be the event that the em- ployee does not have a graduate degree. Find the value of P ( B | A ) . (d) Are the events A and B in part (c) independent? Justify your answer. (e) Two employees are selected randomly to attend a lunch with the CEO of the company. What is the probability they both have a graduate degree? Solution. The easiest way to analyze this problem is to write down a table summarizing the information which we know. We’ll use two columns, one denoted G (for those with graduate degrees) and one denoted G (for those without graduate degrees; and two rows, one labeled R (for ‘Rich’, er, those making $100,000 or more) and one labeled R (for those making less than $100,000). We’ll mark the row sums to the right of the respective rows, and the column sums just below the respective columns. In the lower right column we enter 200, which should be both the row sum and the column sum for that column. Here’s the result: G G Total R 32 R 143 Total 47 200 The first thing we notice is that in order for the last row to sum to 200, the column sum for G must be 153; and in order for the last column to sum to 200, the entry opposite R must be 168. So here’s what we know: G G Total R 32 R 143 168 Total 47 153 200 But this immediately allows us to fill in the R G entry (25 = 168 143) and the R G entry (10 = 153 143): G G Total R 10 32 R 25 143 168 Total 47 153 200 And finally we fill in the R G entry either from the row (22 = 32 10) or from the column (22 = 47 25): G G Total R 22 10 32 R 25 143 168 Total 47 153 200 Now we’re ready to answer the questions. (a) “Makes less than $100,000” is R ; “has a graduate degree” is G ; there are 25 people in R G , as we can see from the final filled-in table, so the probability of randomly choosing one of these 25 from the 200 total employees is 25 / 200, or 0 . 125. (b) “Makes at least $100,000” is R ; “has a graduate degree” is G . The union of column G and row R contains 22 + 10 + 25 = 57 individuals.
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