Questions / Quant trading, Information & Bayes
Separate joint and conditional probabilities
Market regime H occurs 30% of days. A signal fires on 60% of H days and 20% of other days. What fraction of all days have a signal?
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Show the answer and worked solution
Answer: 32% (8/25)
- Split all days into regime H (30%) and its complement (70%).
- Signal-day mass from H is 0.3 × 0.6 = 0.18; signal-day mass from other regimes is 0.7 × 0.2 = 0.14.
- The total is 0.18 + 0.14 = 0.32 of all days, which equals 32%. Both 0.32 and 32% represent the same answer.
Interview shortcut. Total probability: 0.3 × 0.6 + 0.7 × 0.2.
Takeaway. The law of total probability weights each conditional rate by its group frequency.
In the app this has a follow-up where one thing about the setup changes. Try the follow-up.