Probability Review Summary

Statistics · Axiom Academy

Let's review the essential probability concepts for statistics. Probability Range: For any event A, 0 ≤ P(A) ≤ 1 Complement Rule: P(A') = 1 - P(A), where A' is "not A" Addition Rule: P(A or B) = P(A) + P(B) - P(A and B) Mutually Exclusive Events: If A and B cannot occur together, then P(A or B) = P(A) + P(B) Independent Events: P(A and B) = P(A) × P(B) when A and B are independent Definition: P(A|B) = P(A and B) / P(B), the probability of A given B occurred Multiplication Rule: P(A and B) = P(A|B) × P(B) Independence Test: Events are independent if P(A|B) = P(A) Bayes' Theorem: P(A|B) = [P(B|A) × P(A)] / P(B) Law of Total Probability: P(B) = Σ P(B|Aᵢ) × P(Aᵢ) for partitions of sample space Example Recap: Calculating Probability with a Standard Deck Step 1 - Identify the Sample Space: A standard deck has 52 cards total Step 2 - Count Favorable Outcomes: For drawing a heart, there are 13 hearts in the deck Step 3 - Apply the Formula: P(Heart) = 13/52 = 1/4 = 0.25 Step 4 - Interpret the Result: There is a 25% chance of drawing a heart, which makes sense since hearts are 1 of 4 suits Discrete Uniform: All outcomes equally likely (e.g., fair die roll) Binomial: n independent trials with probability p of success: P(X=k) = C(n,k) × p^k × (1-p)^(n-k) Poisson: Counts events in fixed interval: P(X=k) = (λ^k × e^(-λ)) / k! Normal (Gaussian): Bell-shaped, continuous distribution with mean μ and standard deviation σ Exponential: Time between events in Poisson process

This is the written version of the interactive lesson above. See the full Statistics course.