Probability Rules to Know for AP Statistics

Understanding probability rules is key in statistics. These rules help us calculate the likelihood of events happening, whether they can occur together or not. Mastering these concepts lays the groundwork for analyzing data and making informed decisions.

1. Addition Rule for Mutually Exclusive Events

   • States that if two events cannot occur at the same time, the probability of either event occurring is the sum of their individual probabilities.
   • Formula: P(A or B) = P(A) + P(B).
   • Useful for calculating probabilities in scenarios where events are distinct and do not overlap.

2. Addition Rule for Non-Mutually Exclusive Events

   • Applies when two events can occur simultaneously, requiring adjustment for their overlap.
   • Formula: P(A or B) = P(A) + P(B) - P(A and B).
   • Important for accurately determining probabilities in situations where events share outcomes.

3. Multiplication Rule for Independent Events

   • Used when the occurrence of one event does not affect the occurrence of another.
   • Formula: P(A and B) = P(A) * P(B).
   • Essential for calculating joint probabilities in experiments with independent trials.

4. Multiplication Rule for Dependent Events

   • Applies when the outcome of one event influences the outcome of another.
   • Formula: P(A and B) = P(A) * P(B|A), where P(B|A) is the conditional probability of B given A.
   • Critical for understanding sequences of events where prior outcomes affect future probabilities.

5. Complement Rule

   • States that the probability of an event not occurring is equal to one minus the probability of the event occurring.
   • Formula: P(A') = 1 - P(A).
   • Useful for simplifying calculations, especially when dealing with "at least one" scenarios.

6. Law of Total Probability

   • Provides a way to calculate the total probability of an event based on a partition of the sample space.
   • Formula: P(A) = Σ P(A|B_i) * P(B_i), where B_i are mutually exclusive events that cover the entire sample space.
   • Important for breaking down complex probability problems into manageable parts.

7. Bayes' Theorem

   • A method for finding conditional probabilities, allowing for the updating of probabilities based on new evidence.
   • Formula: P(A|B) = [P(B|A) * P(A)] / P(B).
   • Essential for decision-making processes in uncertain conditions, particularly in fields like medicine and finance.

8. Conditional Probability

   • Refers to the probability of an event occurring given that another event has already occurred.
   • Formula: P(A|B) = P(A and B) / P(B).
   • Key for understanding relationships between events and for calculating probabilities in dependent scenarios.

9. Probability of At Least One Event

   • Calculates the probability of at least one of several events occurring, often using the complement rule.
   • Formula: P(at least one A) = 1 - P(none of A).
   • Useful in scenarios where multiple trials or events are considered, such as in games or risk assessments.

10. Permutations and Combinations in Probability

    • Permutations refer to the arrangement of items where order matters, while combinations refer to selections where order does not matter.
    • Formulas: Permutations: nPr = n! / (n-r)!, Combinations: nCr = n! / [r!(n-r)!].
    • Important for calculating probabilities in scenarios involving selections or arrangements of items.