Grades 9–12Probability & StatisticsPlanned
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
The lesson for this topic has not been written yet. The outline below is part of the curriculum map and has not been reviewed.
Builds on
Required prerequisites, all the way down: 18
You will learn to
- 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.
Definition
- 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 = ∅.
Theorem
- 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.
Common misconceptions
- 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'.
Leads to
Optional discoveries
- Statistical inference Bayesian statistics uses Bayes' theorem to update beliefs about unknown quantities in the light of data.