Probability
Probability in the PUC paper moves from conditional probability to Bayes' theorem and then to random variables. You must compute P(A|B), decide whether events are independent, apply the theorem of total probability, and work with a probability distribution, its mean and variance, including the binomial distribution.
Rules
- P(A|B) = P(A ∩ B)/P(B) for P(B) > 0; multiplication rule P(A ∩ B) = P(A)P(B|A).
- Independent events: P(A ∩ B) = P(A)P(B), equivalently P(A|B) = P(A). Independence is not the same as mutual exclusivity.
- Total probability: for a partition E₁, …, Eₙ, P(A) = Σ P(Eᵢ)P(A|Eᵢ).
- Bayes: P(Eᵢ|A) = P(Eᵢ)P(A|Eᵢ) / Σ P(Eⱼ)P(A|Eⱼ).
- Random variable: mean μ = Σ xᵢpᵢ, variance = Σ xᵢ²pᵢ − μ², with Σ pᵢ = 1.
- Binomial: P(X = r) = ⁿCᵣ pʳ qⁿ⁻ʳ, mean np, variance npq.
Worked example
A fair die is thrown 4 times. Find the probability of exactly two sixes. With n = 4, p = 1/6, q = 5/6 and r = 2: P(X = 2) = ⁴C₂(1/6)²(5/6)² = 6 · 25/1296 = 25/216. The distribution has mean np = 2/3 and variance npq = 5/9.
Traps
- Swapping P(A|B) and P(B|A) in Bayes problems.
- Assuming independence from P(A ∩ B) without checking the product.
- Using P(A ∪ B) = P(A) + P(B) and forgetting to subtract P(A ∩ B).
- Writing a distribution whose probabilities do not sum to 1.