Problem Set #3 (Chapter 4)

27) Students at Mideastern University. According to the National Center for Education Statistics (NCES), nearly 20% of the bachelor’s degrees awarded in 2019 were business degrees (NCES website). Suppose that 24% of students at Mideastern University study business. Students at Mideastern University either live on campus or commute to campus. It is known that 38% of students commute to campus at Mideastern University and 59.5% of students are either business students or live on campus.

a) What is the probability that a randomly selected student at Mideastern University lives on campus?

b) What is the probability that a randomly selected student at Mideastern University studies business and lives on campus?

c) Is it true that a student studying business and a student commuting to campus at Mideastern University are mutually exclusive events? Explain.

29) Ivy League Admissions. Highschool seniors with strong academic records apply to the nation’s most selective colleges in greater numbers each year. Because the number of slots remains relatively stable, some colleges reject more early applicants. Suppose that for a recent admissions class, an Ivy League college received 2851 applications for early admission. Of this group, it admitted 1033 students early, rejected 854 outright, and deferred 964 to the regular admission pool for further consideration. In the past, this school has admitted 18% of the deferred early admission applicants during the regular admission process. Counting the students admitted early and the students admitted during the regular admission process, the total class size was 2375. Let E, R, and D represent the events that a student who applies for early admission is admitted early, rejected outright, or deferred to the regular admissions pool.

a) Use the data to estimate P(E), P(R), and P(D).

b) Are events E and D mutually exclusive? Find P(E ∩ D).

c) For the 2375 students who were admitted, what is the probability that a randomly selected student was accepted during early admission?

d) Suppose a student applies for early admission. What is the probability that the student will be admitted for early admission or be deferred and later admitted during the regular admission process?

33) Intent to Pursue MBA. Students taking the Graduate Management Admissions Test (GMAT) were asked about their undergraduate major and intent to pursue their MBA as a full-time or part-time student. A summary of their responses follows.

a) Develop a joint probability table for these data.

b) Use the marginal probabilities of undergraduate major (business, engineering, or other) to comment on which undergraduate major produces the most potential MBA students.

c) If a student intends to attend classes full-time in pursuit of an MBA degree, what is the probability that the student was an undergraduate engineering major?

d) If a student was an undergraduate business major, what is the probability that the student intends to attend classes full-time in pursuit of an MBA degree?

e) Let A denote the event that the student intends to attend classes full-time in pursuit of an MBA degree, and let B denote the event that the student was an undergraduate business major. Are events A and B independent? Justify your answer.

34) On-Time Performance of Airlines. The Bureau of Transportation Statistics reports on-time performance for airlines at major U.S. airports. JetBlue, United, and US Airways share terminal C at Boston’s Logan Airport. Suppose that the percentage of on-time flights reported was 76.8% for JetBlue, 71.5% for United, and 82.2% for US Airways. Assume that 30% of the flights arriving at terminal C are JetBlue flights, 32% are United flights, and 38% are US Airways flights.

a) Develop a joint probability table with three rows (the airlines) and two columns (on-time and late).

b) An announcement is made that Flight 1382 will be arriving at gate 20 of terminal C. What is the probability that Flight 1382 will arrive on time?

c) What is the most likely airline for Flight 1382? What is the probability that Flight 1382 is by this airline?

d) Suppose that an announcement is made saying that Flight 1382 will now be arriving late. What is the most likely airline for this flight? What is the probability that Flight 1382 is by this airline?

42) Credit Card Defaults. A local bank reviewed its credit card policy with the intention of recalling some of its credit cards. In the past approximately 5% of cardholders defaulted, leaving the bank unable to collect the outstanding balance. Hence, management established a prior probability of 0.05 that any particular cardholder will default. The bank also found that the probability of missing a monthly payment is 0.20 for customers who do not default. Of course, the probability of missing a monthly payment for those who default is 1.

a) Given that a customer missed one or more monthly payments, compute the posterior probability that the customer will default.

b) The bank would like to recall its card if the probability that a customer will default is greater than 0.20. Should the bank recall its card if the customer misses a monthly payment? Why or why not?

43) Prostate Cancer Screening. According to a 2018 article in Esquire magazine, approximately 70% of males overage 70 will develop cancerous cells in their prostate. Prostate cancer is second only to skin cancer as the most common form. of cancer for males in the United States. One of the most common tests for the detection of prostate cancer is the prostate-specific antigen (PSA) test. However, this testis known to have a high false-positive rate (tests that come back positive for cancer when no cancer is present). Suppose there is a 0.02 probability that a male patient has prostate cancer before testing. The probability of a false-positive testis 0.75, and the probability of a false-negative (no indication of cancer when cancer is actually present) is 0.20.

a) What is the probability that the male patient has prostate cancer if the PSA test comes back positive?

b) What is the probability that the male patient has prostate cancer if the PSA test comes back negative?

c) For older males, the prior probability of having cancer increases. Suppose that the prior probability of the male patient is 0.30 rather than 0.02. What is the probability that the male patient has prostate cancer if the PSA test comes back positive? What is the probability that the male patient has prostate cancer if the PSA test comes back negative?

d) What can you infer about the PSA test from the results of parts (a), (b), and (c)?