Foundations Summary

Probability · Axiom Academy

SUMMARY Probability Foundations Let's review the essential concepts of sample spaces, events, and counting methods. Sample Space (S): The set of all possible outcomes of an experiment Event (A): A subset of the sample space representing outcomes of interest Complement ( ): All outcomes not in event A Union ( ): Outcomes in A or B or both Intersection ( ): Outcomes in both A and B Definition: Probability is the ratio of favorable outcomes to total outcomes Range: All probabilities are between 0 and 1 (or 0% to 100%) Addition Rule: For any two events, subtract the overlap to avoid double-counting Mutually Exclusive: Events that cannot happen together have no overlap This formula works when all outcomes in the sample space are equally likely . Example Recap: Selecting a Committee Step 1 - Identify the question: How many ways can we select 3 people from a group of 10 for a committee? Step 2 - Determine if order matters: In a committee, order doesn't matter (Alice-Bob-Carol is the same as Carol-Bob-Alice), so we use combinations Step 3 - Apply the formula: Use When to Use Each Counting Method Multiplication Principle: Use when making multiple sequential choices with replacement (like passwords with repeated digits) Permutations P(n,r): Use when order matters and repetition is not allowed (race finishing positions) Combinations C(n,r): Use when order doesn't matter and repetition is not allowed (selecting committee members)

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