04 Jun Question Instructions: Please answer all questions clearly and
Question
Instructions: Please answer all questions clearly and completely. Justify and support your conclusions. If you use graphs or tables in your answers, they must be clear enough that I can understand them (this means labeling axes, variables, and so forth). If a question requires you to make calculations, you must show your work. Your file should be ready as if you were going to print it and hand in a hard copy. If you make calculations using Excel, please include your spreadsheet output as an appendix and submit it via ANGEL. Each of the following questions is worth ten (10) points.
Question 1: 5 fair coins are flipped. (A fair coin means that P(Heads) = 0.5 and P(Tails) = 0.5)). What is the probability that 3 of the coins show Heads? You may assume that the coin flips are independent.
Question 2: Of a group of students, 40% are from Pennsylvania,10% are Chemistry majors, and 3% are both. If a student is picked at random, what is the probability that he/she is neither from PA nor a Chem major? What is the probability that a student is a Chem major, given that he/she is from PA? Are from PA and Chem major independent events?
Questions 3 and 4 refer to the disease problem in the Strogatz article (on ANGEL): The probability that a person has a disease is 0.8 percent. If a person has the disease, the probability is 90 percent that he/she will have a positive test result for the disease. If a person does not have the disease, the probability is 7 percent that he/she will still have a positive test result.
Question 3: Imagine a person with a negative test result. Calculate the probability that this
person has the disease using the summary table on slide 37 or the probability tree on slide 43
of the Introduction to Probability lecture notes.
Question 4: Repeat Question 3, but use Bayes Rule explicitly. Verify that you get the same
answer as in Question 3.
Questions 5 through 10 refer to the following variation on the Strogatz problem: A new test has
been devised for detecting a particular disease. If the test is applied to a person who has this
disease, the probability of a positive test result (i.e., the test states that they have the disease) is
0.95 and the probability of a negative test result is 0.05. If the test is applied to a person who
does not have this particular disease, the probability of a positive test result is 0.08. Suppose
that one person in 10,000 has this disease.
For this set of questions, it will be useful to define the following:
D is the event that a randomly chosen person has the disease, with P(D) = 0.0001.
ND is the event that a randomly chosen person does not have the disease, with
P(ND) = 0.9999.
S is the event that the test comes out positive, with P(S|D) = 0.95 and P(S|ND) = 0.08.
NS is the event that the test comes out negative, with P(NS|D) = 0.05 and P(NS|ND) =
0.92.
Question 5: If a person selected at random tests positive, what is the probability that this person
has this disease?
Question 6: If one person in 20 has this disease (instead of 1 in 10,000), how would your answer
to Question 5 change?
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