题目
A rare disease affects 1 in 10,000 people. A screening test has 99% sensitivity (true positive rate) and 99% specificity (true negative rate). (a) A randomly chosen person tests positive. What is the probability they actually have the disease? (b) The same person takes a second independent test (same sensitivity and specificity) and tests positive again. Now what is the probability they have the disease? (c) A hospital administrator sees that the test is '99% accurate' and proposes mandatory screening of all 100,000 employees, arguing that 'almost everyone who tests positive will be sick.' Quantify how many of the expected positive results will be false positives, and explain why the administrator's reasoning is flawed.
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In algorithmic trading, a signal fires on 0.1% of days and predicts a profitable trade with 95% precision when the signal is real. However, on the 99.9% of days when there is no real signal, noise triggers the same alert 2% of the time. What fraction of all alerts correspond to real signals? How should the trading system handle this?
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