Ch 13 covers advanced probability: conditional probability, multiplication theorem, independent events, Bayes' theorem, random variables, probability distributions, and the binomial distribution.
Conditional probability: P(A|B) = P(A∩B)/P(B). Multiplication theorem: P(A∩B) = P(A)·P(B|A). Events A, B are independent if P(A∩B) = P(A)·P(B), i.e., P(A|B) = P(A).
If E₁,E₂,…,Eₙ are mutually exclusive and exhaustive events: P(Eᵢ|A) = P(Eᵢ)·P(A|Eᵢ) / Σⱼ P(Eⱼ)·P(A|Eⱼ). Used when we need to "reverse" conditional probabilities.
A random variable X maps outcomes to numbers. P(X = xᵢ) = pᵢ. Mean E(X) = Σxᵢpᵢ. Variance = E(X²) − [E(X)]². Binomial: P(X=r) = nCr pʳ qⁿ⁻ʳ, Mean = np, Variance = npq.
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Bayes' theorem calculates the probability of a cause given an observed effect. For example: given a positive medical test result, what is the probability the person actually has the disease? It updates prior probabilities using new evidence.
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