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The Lumenary
الصفوف 9–12Probability & Statisticsمُخطَّط له

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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ProbabilityCombinatorics

عدد المتطلبات السابقة الإلزامية (حتى الأساس): 18

ستتعلّم

  • 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.

تعريف

  • 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 = ∅.

مبرهنة

  • 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.

مفاهيم خاطئة شائعة

  • 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'.

يقود إلى

Probability theory

اكتشافات اختيارية

  • Statistical inference Bayesian statistics uses Bayes' theorem to update beliefs about unknown quantities in the light of data.

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