Math & Statistics · 概率 · Updated 2026-10-25
What Is Probability?
Definition
Probability lives in [0,1]. Mutually exclusive events add; independent events multiply; conditional probability P(A|B) grounds Bayes’ rule. Watch the base-rate fallacy: a 99% accurate test with 0.1% prevalence yields <9% true positives among the flagged.
Formula
P(A∩B)=P(A)P(B); Bayes: P(A|B)=P(B|A)P(A)/P(B)
Worked Example
After five heads, the next flip is still 1/2 (gambler’s fallacy). P(at least one double-six in 25 rolls) ≈ 50.6%.
Common Misconception
"Red came up 5 times, so black is due" - independent events have no memory; the gambler's fallacy is the casino's favorite illusion. "The test is 99% accurate so a positive means disease" - with low base rates, false positives dominate.
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