A signal is not a posterior
Ten percent of datasets contain a signal. A filter retains 80% of signal datasets and 20% of nonsignal datasets. What fraction of retained datasets truly contain a signal?
Start with 1,000 datasets
Of 100 signal datasets, 80 survive. Of 900 nonsignal datasets, 180 survive. There are 260 retained datasets and 80 true signals. The answer is 80/260 = 4/13 ≈ 30.77%, not 80%.
Fast method
Use ps/[ps+(1−p)f]. The numerator is retained signal mass; the denominator is all retained mass. Division renormalizes the selected population. A frequency table often beats memorizing symbols.
Change one assumption
If nonsignal retention falls to 5%, the posterior becomes 0.08/(0.08+0.045)=0.64. Specificity matters enormously for rare signals. Without a base-rate assumption there is no unique posterior.
Follow-up: two filters agree
You cannot multiply likelihoods unless outputs are conditionally independent given the state. Correlated evidence is an easy route to unjustified confidence.
Jane Street’s public guide establishes probability and expectation as preparation topics. This example is original QuantPrep material, not an attributed employer question.
Try a fresh question
This optional exercise uses the actual parameterized bank.
Worked method
Keep the practice connected
Train Probability · Follow the Quant Path · Evidence standard