返回故事列表Probability Paradox

99%準確率檢測悖論：為什麼不懂貝氏定理連醫生與AI工程師也會誤判？

If a test with 99% sensitivity and 99% specificity flags you positive for a 1-in-1,000 disease, what is your true probability? Only 9%.

P(D \mid +) = \frac{P(+\mid D)P(D)}{P(+\mid D)P(D) + P(+\mid D^c)P(D^c)} \approx 9.0\%

Base Rate Fallacy: When prior P(D) is tiny, false positives from the vast healthy majority outnumber true positives.

Sequential Bayesian Updating: A second independent positive test uses the first posterior (9%) as its new prior, jumping confidence above 90%.

Crucial for fraud detection, autonomous driving sensor fusion, and medical diagnostic calibration.

⚡ 本文涉及的核心數學公式

#P02條件概率與乘法定理 #P03貝氏定理 (後驗概率推斷)#P05二項分布與伯努利試驗 #P09香農資訊熵與交叉熵損失 (Cross-Entropy)

1. Visualizing with 10,000 People

\text{True Positives} = 10 \times 0.99 \approx 10, \quad \text{False Positives} = 9990 \times 0.01 \approx 100

Out of 10,000 people, 10 have the disease (producing ~10 true positives), while 9,990 are healthy (producing ~100 false positives at a 1% false-positive rate). Thus only 10 of 110 positives (~9%) are truly sick.

2. Sequential Bayesian Updating

P(D \mid +_1, +_2) = \frac{0.99 \times 0.0909}{0.99 \times 0.0909 + 0.01 \times 0.9091} \approx 90.8\%

Feeding the 9.09% posterior as the new prior into a second independent test boosts the posterior probability to 90.8%. This iterative belief update is how Kalman filters track autonomous vehicles.