Round 3: Stats & Probability (Explain It)

11. State Bayes' theorem and work through: 1% disease rate, 95% sensitivity, 3% false positive. Person tests positive.

P(D|+) = (0.95 × 0.01) / (0.95 × 0.01 + 0.03 × 0.99) = 0.0095 / (0.0095 + 0.0297) = 0.0095 / 0.0392 ≈ 24.2%

12. What does p = 0.04 mean? If H₀ were true (no effect), there's only a 4% chance of observing data this extreme. At α=0.05, we reject H₀.

13. Type I vs Type II? Type I (α) = false positive, crying wolf. Type II (β) = false negative, missing the wolf. Power = 1-β ≥ 0.80.

14. Why always switch in Monty Hall? Initial pick = 1/3. Other two doors = 2/3. Host reveals one is wrong, remaining door gets the full 2/3. Switching = 2/3 win rate.

15. 95% confidence interval: what does it ACTUALLY mean? If you repeated the experiment 100 times, ~95 of the intervals would contain the true value. NOT "95% probability the true value is in this specific interval."