A semiconductor testing bin contains 3 visually identical ASICs. Two are standard and operate correctly. One is a defective "stuck-at-1" chip that outputs a logic '1' exactly 100% of the time. A standard chip outputs '1' 50% of the time and '0' 50% of the time on a random test vector. You pick one chip at random, test it once, and it outputs a '1'. What is the probability that you picked the defective chip?
Explanation: Using Bayes' Theorem: P(Defective) = 1/3, P(Standard) = 2/3. P(Output=1 | Defective) = 1.0. P(Output=1 | Standard) = 0.5. We want P(Defective | Output=1) = [P(Output=1 | Defective) P(Defective)] / P(Output=1). Total P(Output=1) = (1.0 1/3) + (0.5 * 2/3) = 1/3 + 1/3 = 2/3. Therefore, P(Defective | Output=1) = (1/3) / (2/3) = 1/2.
