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Problem Set 1

Problem 1 (1.12). Consider the probability space (Ω, F, P) for this problem.

(a) State the three axioms of probability theory and explain in a sentence the significance of each.

(b) Derive the following formula, justifying each step by reference to the appropriate axiom above,

P{A ∪ B} = P{A} + P{B} − P{A ∩ B}

where A and B are arbitrary events in the field F.

Problem 2 (1.28). A fair die is tossed three times. Given that a 2 appears on the first toss, what is the probability of obtaining the sum 7 on the three tosses?

Problem 3 (1.30). The following problem was given to 60 students and doctors at the famous Hevardi Medical School (HMS): Assume there exists a test to detect a disease, say D, whose prevalence is 0.001, that is, the probability, P{D}, that a person picked at random is suffering from D, is 0.001. The test has a false-positive rate of 0.005 and a correct detection rate of 1. The correct detection rate is the probability that if you have D, the test will say that you have D. Given that you test positive for D, what is the probability that you actually have it? Many of the HMS experts answered 0.95 and the average answer was 0.56. Show that your knowledge of probability is greater than that of the HMS experts by getting the right answer of 0.17.

Problem 4 (1.33). A large class in probability theory is taking a multiple-choice test. For a particular question on the test, the fraction of examinees who know the answer is p; 1 − p is the fraction that will guess. The probability of answering a question correctly is unity for an examinee who knows the answer and 1/m for a guessee; m is the number of multiple-choice alternatives. Compute the probability that an examinee knew the answer to a question given that he or she has correctly answered it.

Problem 5 (1.49). A smuggler, trying to pass himself off as a glass-bead importer, attempts to smuggle diamonds by mixing diamond beads among glass beads in the proportion of one diamond bead per 1000 beads. A harried customs inspector examines a sample of 100 beads. What is the probability that the smuggler will be caught, that is, that there will be at least one diamond bead in the sample?

Problem 6 (1.55). An automatic breathing apparatus (B) used in anesthesia fails with probability PB. A failure means death to the patient unless a monitor system (M) detects the failure and alerts the physician. The monitor system fails with probability PM. The failures of the system components are independent events. Professor X, an M.D. at Hevardi Medical School, argues that if PM > PB installation of M is useless.1 Show that Prof. X needs to take a course on probability theory by computing the probability of a patient dying with and without the monitor system in place. Take PM = 0.1 = 2PB.

Problem 7 (1.66). Let us assume that two people have their birthdays on the same day if both the month and the day are the same for each (not necessarily the year). How many people would you need to have in a room before the probability is 1/2 or greater that at least two people have their birthdays on the same day?