(a) Establish all possible relationships among the four probabilities P1, 91, 92 and p2.

(b) What is the probability that any given test is Positive, if it is known that the system is Positive? 

(c) What is the probability that a Positive test is in fact indicative of a Positive status of the system? For this question, answer the question assuming a prevalence of 50% for a positive test, and a prevalence of 10%. 

(d) What is the probability that a Negative test is in fact indicative of a Negative status of the system? For this question, answer the question assuming a prevalence of 50% for a positive test, and a prevalence of 10%.

Specificity and Sensitivity of a test A test for the Positive/Negative status of a binary experiment or system consists of predicting whether the system is in a Positive or a Negative state. The experiment should be designed in such a way as to minimize the rate of occurrence of False Negatives (FN) and False Positives (FP), defined respectively as a test that returns a Negative result when the system was known to be Positive, and a test that returns a Positive result when the system was known to be Negative. Estimating the rate of FN and FP relies on the knowledge of the true status of the system. The four possible combinations of the true and predicted status of the system are: 1. 

True Positive (TP): system is Positive, and the test returns a Positive value 2. False Negative (FN): system is Positive, but the test returns a Negative value 

3. False Positive (FP): system is Negative, but test returns a Positive value 4. 

True Negative (TN): system is Negative, and the test returns a Negative value Assume that the test is calibrated, e. g., via clinical studies, in such a way that the following probabilites have been estimated: 1. 

Pı: Probability of TP, or P(test=P/status=P) 2. q1: Probability of FN, or P(test=N/status=P) 3. 42: Probability of FP, or P(test=P/status=N) 4. P2: Probability of TN, or P(test=N/status=N). There are two key concepts to assess the usefulness or the quality of a test. One is that of specificity, defined as ТР sensitivity TP+FN and the other is the specificity, defined as specificity = TN TN+FP' The sensitivity measure the rate of true positives, relative to all cases known to be positive (i.e., true positives and false negatives). 

The specificity focuses on a simular measure of the rate of true negtive tests. Neither of these two quantities answer the ultimate question of what is the probability that a Positive (or Negative) test is truly indicative of a Positive (or Negative) status. Such probability can be phrased as P(status=P/test=P), or P(status=N/test=N), i.e., it is a posterior probability after the result of the test is known. Bayes' theorem is useful to answer this question, since it provides the means to invert the conditioning of probabilities. Bayes' theorem relies on the knowledge of prior probabilities that a test returns a positive or a negative test. 

These prior probabilities can be determined experimentally using the prevalence of a condition. For example, if it is known that 50% of a population is expected to be positive to a disease, one can use a prior probability of p = 0.5 for the test returning a positive result. The sensitivity and specificity are independent of the prevalence, but the posterior probabilites will depend on it. Using the theory provided above,

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