Question

3. Let C be the event that a patient suffers from a certain condition, and let T denote a positive result from a lab test that is designed to detect the presence of said condi- tion. Suppose that the proportion of the population that actually has the condition IS E E (0,1). Additionally, suppose that, when the condition is actually present in a patient, the test is positive with probability a (0,1). On the other hand, when the patient does not suffer from the condition, the test will yield a false-positive result witlh probability β E (0, 1). We assume the test is "well-behaved", in the sense that α > β. (a) Show that the probability of suffering from the disease given a positive test out- come is given by the expression CYE Assuming a, B fixed, what is the behavior of this probability when the condition is extremely rare in the population (i.e., e - 0)? (b) Suppose that a patient takes the test K times, KEN. Let Tk the event where the k-th test yields a positive outcome, k E{1,... , K}. Assume that, conditional on C, T, ..., TK) are independent. Show that the probability of suffering from the condition given that all K tests show a positive result is: Assuming all else fixed, what is the behavior of this probability when K -> oo? (c) Let є_1/10000, d-0.999, B-0.01: i. What is the value of P(CT)? ii. Find inf K EN P(CITi nnTK0.99. In other words, what is the least amount of positive tests one needs to observe to be at least 99% confident that one suffers from the condition?

Answer 1

(a) According to the problem. PTIC)-a. PTIC")-β, PC) P(TC)P(C) P(CIT) 8(1- p Qr

(b) P(TinT n . T C) = P(T C )PlT C) P(T C ) (since T, c's are independent) PlTi nT2n nTkļC c)-PlT |C")Pl T2IC ) PlTk |C") (since T, |C's are independent) = ßk (since P(T,IC) = β y i = 1, 2, , k) 1 since K oc and α > β > 0 a. () ) P(CT)01 0.0099 0.01 (1 1/10000)/(0.999/10000)

(ii) 9999,-ー0.01/0.999~ 0.01 > 0.99 or, 1 + (0.01)/( * 9999 < 0.99 1(0.01)k 9999 (0.01)K * 99996 0.01 or, (0.01)K-1 < 9999 or, or, K> -log 9999 2. log 0.01