Probability Problem Solving Guide

Probability · Axiom Academy

SUMMARY Probability Problem Solving Guide Master strategies, decision frameworks, and techniques for tackling any probability problem Distribution Selection Decision Tree Permutations (order matters): Arranging n items in order: Combinations (order doesn't matter): Choosing r from n: Key question: Does rearranging give a different outcome? With replacement: Use multiplication principle Conditional Probability Problems Tree diagrams: Draw when multiple stages/events occur sequentially Bayes' Theorem: For "reverse" conditional probabilities Law of Total Probability: When outcome can happen via multiple paths E(X): Discrete: , Continuous: integrate Distribution Identification from Words "Number of successes in n trials": Binomial "Number of events per hour/area": Poisson "Time until first occurrence": Exponential/Geometric "Average of many samples": Normal (CLT) Universal Problem-Solving Framework Identify the random variable: Define X clearly. What are you counting/measuring? Classify the problem type: Is this about a single value P(X = k), a range P(X ≤ k), expected value E(X), or conditional probability? Choose the distribution: Use the decision tree above to identify which distribution applies Extract parameters: Find n, p, λ, μ, σ from the problem statement Set up the calculation: Write the formula with your specific values Compute: Use tables, calculator, or normal approximation as appropriate Interpret: State your answer in context with proper units

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