MATHSTAT318_HW4_Solutions

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Pennsylvania State University *

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318

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Feb 20, 2024

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1 MATH/STAT 318 – Homework 4 HW4 is due by 11:59 PM on Thursday, February 8. Instructions Some details: 1. When submitting homework in Canvas, multiple pages must be scanned and saved as a single PDF document. If you submit other file types, such as .jpeg, .png, .heic, etc., instead of a .pdf, will result in points being deducted or a score of 0. 2. Not knowing how to scan your homework assignment into a PDF document is not a valid excuse for late submissions. If you have never scanned your homework previously, try scanning a few sheets of paper just to check whether you are able to do it. 3. Failure to upload your homework assignment OR uploading the incorrect assignment will result in a grade of 0 for the assignment. It is each student's responsibility to check that the correct homework assignment is uploaded before the due date! 4. It is each student’s responsibility to make sure that the file you submit is complete, the problems are in the correct order, and the scan is readable. 5. The Instructor and TA must be able to easily read all uploaded homework. Messy and hard-to-read Canvas submissions will have points deducted or receive a score of 0. 6. While completing the homework problems, show all relevant work. Credit is given mostly for the method/work, not the answer. 7. You are encouraged to work with other students, but each student must write and submit their own assignment. Even if you have worked with others, you should not be submitting identical assignments. 8. For the policy of late assignments, see the syllabus. The rest of this document is formatted like an exam, roughly one page per problem. You might want to consider printing this document. I would suggest doing the problems on scratch paper first. Once you have found a satisfactory solution, rewrite the solution neatly on this document. When appropriate, please circle your final answer. At the conclusion of the Homework 4, you should upload a single pdf file to Canvas containing your solutions. NOTE: Your work matters as much, if not more, than just writing the final answer. Rather than simply writing a final numerical answer, be sure to include sufficient work to communicate how you are arriving at the answer.
2 Problem 1 (6 Points) It is known that 30% of a certain company’s washing machines require service while under warranty, whereas only 10% of its dryers need such service. If someone purchases both a washer and a dryer made by this company, find the following. a. What is the probability that both machines will need service under warranty? b. What is the probability that neither machine will need service under warranty?
3 Problem 2 (6 Points) (Problem 1.4-2 in textbook.) Suppose that ࠵?(࠵?) = 0.3 and ࠵?(࠵?) = 0.6 . a. Find ࠵?(࠵? ∪ ࠵? ) when ࠵? and ࠵? are independent. b. Find ࠵?(࠵? ∣ ࠵? ) when ࠵? and ࠵? are mutually exclusive.
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4 Problem 3 (10 Points) Suppose that a student in the class lives on campus with probability 0.1. Five students are randomly selected in the class. a. What is the probability that none of them live on campus? b. What is the probability that at least one student lives on campus? c. What is the probability that only the first student selected out of the five lives on campus? d. What is the probability that exactly one student lives on campus?
5 Problem 4 (5 Points) (Similar to 1.4-8 in textbook but calling the dice 1, 2, and 3 instead of A, B, and C.) Die 1 has orange on one face and blue on five faces. Die 2 has orange on two faces and blue on four faces. Die 3 has orange on three faces and blue on three faces. All are fair dice meaning that each of the six sides on each die are equally likely. If the three dice are rolled, find the probability that exactly two of the three dice produce orange.
6 Problem 5 (6 Points) (Problem 1.5-2 from textbook) Bean seeds from supplier A have an 85% germination rate and those from supplier B have a 75% germination rate. A seed-packaging company purchases 40% of its bean seeds from supplier A and 60% from supplier B and mixes these seeds together. a. Find the probability that a seed selected at random from the mixed seeds will germinate. b. Given that a seed germinates, find the probability that the seed was purchased from supplier A.
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7 Problem 6 (5 Points) Only 1 in 1000 adults is afflicted with a rare disease for which a diagnostic test has been developed. The test is such that when an individual actually has the disease, a positive result will occur 99% of the time, whereas an individual without the disease will show a positive test result only 2% of the time. If a randomly selected individual is tested and the result is positive, what is the probability that the individual has the disease? NOTE: This answer might be smaller than your intuition suggests.
8 Problem 7 (5 Points) An aircraft emergency locator transmitter (ELT) is a device designed to transmit a signal in the case of a crash. The Altigauge Manufacturing Company makes 80% of the ELTs, the Bryant Company makes 15% of them, and the Chartair Company makes the other 5%. The ELTs made by Altigauge have a 4% rate of defects, the Bryant ELTs have a 6% rate of defects, and the Chartair ELTs have a 9% rate of defects (which helps to explain why Chartair has the lowest market share). a. If an ELT is randomly selected from the general population of all ELTs, find the probability that it was made by the Altigauge Manufacturing Company. b. If a randomly selected ELT is then tested and is found to be defective, find the probability that it was made by the Altigauge Manufacturing Company.
9 Problem 8 (7 Points) Flip a fair coin 5 times. Compute the probability of: a. obtaining the sequence ࠵?࠵?࠵?࠵?࠵? . b. obtaining the sequence ࠵?࠵?࠵?࠵?࠵? . c. obtaining at least 3 heads in the 5 trials.
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