九至十二年级Probability & Statistics计划中
Conditional probability and independence
Addition and multiplication rules, conditional probability, independence, Bayes' theorem in simple cases, and probabilities computed by counting.
math.prob.probability-rules
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前置知识
18 项必需的前置知识（一直追溯到底）
你将学会
- Compute conditional probabilities from two-way tables and tree diagrams.
- Decide whether events are independent, and distinguish independence from mutual exclusivity.
- Apply Bayes' theorem to problems such as diagnostic testing with base rates.
定义
- For P(B) > 0, the conditional probability of A given B is P(A | B) = P(A ∩ B)/P(B).
- Events A and B are independent if P(A ∩ B) = P(A)P(B).
- Events A and B are mutually exclusive if they cannot both occur, that is, A ∩ B = ∅.
定理
- P(A ∪ B) = P(A) + P(B) − P(A ∩ B).
- Bayes' theorem: P(A | B) = P(B | A)P(A)/P(B) when P(B) > 0.
- Law of total probability: if B₁, …, Bₙ partition the sample space, P(A) = Σ P(A | Bᵢ)P(Bᵢ).
- Two mutually exclusive events with positive probabilities are never independent.
常见误解
- Confusing P(A | B) with P(B | A) (the prosecutor's fallacy).
- Ignoring base rates when interpreting a positive test result.
- Thinking 'independent' means 'mutually exclusive'.
通向
可选发现
- Statistical inference Bayesian statistics uses Bayes' theorem to update beliefs about unknown quantities in the light of data.