Company A has just developed a diagnostic test for a certain disease. The disease afflicts 1% of the population. The sensitivity of the test is the probability of someone testing positive, given that they have the disease, and the specificity of the test is the probability that of someone testing negative, given that they don't have the disease. Assume that the sensitivity and specificity are both 0.95.
Company B, which is a rival of Company A, offers a competing test for the disease. Company B claims that their test is faster and less expensive to perform than Company A's test, is less painful (Company A's test requires an incision), and yet has a higher overall success rate, where overall success rate is defined as the probability that a random person gets diagnosed correctly.
(a) It turns out that Company B's test can be described and performed very simply: no matter who the patient is, diagnose that they do not have the disease. Check whether Company B's claim about overall success rates is true.
(b) Explain why Company A's test may still be useful.
(c) Company A wants to develop a new test such that the overall success rate is higher than that of Company B's test. If the sensitivity and specificity are equal, how high does the sensitivity have to be to achieve their goal? If (amazingly) they can get the sensitivity equal to 1, how high does the specificity have to be to achieve their goal? If (amazingly) they can get the specificity equal to 1, how high does the sensitivity have to be to achieve their goal?