1. **Problem 65:** A population has 5% men with coronary artery disease. The test accuracy given: \(P(\text{test correct}|\text{disease})=0.24\), \(P(\text{test correct}|\text{no disease})=0.98\). We want \(P(\text{disease}|\text{test positive})\approx ?\) Step 1: Define events: \(D\): has disease, \(\neg D\): no disease \(T\): test positive for disease Given: \[P(D) = 0.05, \quad P(\neg D) = 0.95\] \[P(T|D) = 0.24, \quad P(\neg T|\neg D) = 0.98 \implies P(T|\neg D) = 0.02\] Step 2: Use Bayes' theorem for \(P(D|T)\): \[ P(D|T) = \frac{P(T|D) P(D)}{P(T|D)P(D) + P(T|\neg D)P(\neg D)} \] Step 3: Substitute values: \[ P(D|T) = \frac{0.24 \times 0.05}{0.24 \times 0.05 + 0.02 \times 0.95} = \frac{0.012}{0.012 + 0.019} = \frac{0.012}{0.031} \approx 0.387 \] Step 4: Interpretation: The probability the test is correct when it indicates disease is about 38.7%, closest to option (E) 39%. 2. **Problem 66:** Given topological space \(X = \{ n \in \mathbb{Z} : n \geq 2 \}\) with topology generated by sets \(U_k = \{n \in X : n \text{ divides } k\}\). We are asked for the closure of \(\{n\}\) in \(X\). Step 1: Understanding \(U_k\): Each \(U_k\) includes numbers dividing \(k\). Step 2: The topology's open sets are unions of such \(U_k\). Step 3: A point \(x\) is in the closure of \(\{n\}\) if every open set containing \(x\) intersects \(\{n\}\). Step 4: Every open neighborhood of \(x\) contains some set \(U_k\) with \(x \in U_k\). Since \(x \in U_k\) means \(x | k\), and \(n \in U_k\) means \(n | k\). For open sets containing \(x\) to always intersect \(\{n\}\), \(n\) must divide \(k\) whenever \(x\) divides \(k\). This implies \(x\) divides all multiples of \(n\), so \(x\) divides \(n\). Step 5: Therefore closure of \(\{n\}\) is all \(t \in X\) such that \(t\) divides \(n\). Answer is option (C). **Final answers:** Problem 65: (E) 39% (probability approx 0.387) Problem 66: (C) \(\{t \in X : t \text{ divides } n\}\)