Basic Probability Rules to Know for Intro to Probability

Understanding basic probability rules is key for making informed decisions in business and everyday life. These rules help us calculate the likelihood of events happening, whether they are independent, dependent, or overlapping, guiding us through uncertainty.

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 assessing probabilities in situations where events share common outcomes.

3. Multiplication Rule for Independent Events

   • States that the probability of two independent events both occurring is the product of their individual probabilities.
   • Formula: P(A and B) = P(A) * P(B).
   • Essential for scenarios where the occurrence of one event does not affect the other.

4. Multiplication Rule for Dependent Events

   • Used when the occurrence of one event affects the probability of another event occurring.
   • 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 influence 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 by focusing on the opposite of the event in question.

6. Law of Total Probability

   • Provides a way to calculate the total probability of an event by considering all possible scenarios that could lead to that event.
   • 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 problems into manageable parts.

7. Bayes' Theorem

   • A method for updating the probability of an event based on new evidence or information.
   • Formula: P(A|B) = [P(B|A) * P(A)] / P(B).
   • Essential for decision-making processes that involve conditional probabilities.

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).
   • Important for understanding relationships between events and how they influence each other.

9. Probability of Union of Events

   • Refers to the probability that at least one of several events occurs.
   • Formula: P(A ∪ B) = P(A) + P(B) - P(A ∩ B).
   • Key for assessing the likelihood of multiple events happening together.

10. Probability of Intersection of Events

    • Refers to the probability that both events occur simultaneously.
    • Formula: P(A ∩ B) = P(A) * P(B|A) for dependent events or P(A) * P(B) for independent events.
    • Crucial for understanding the overlap between events and their joint occurrence.