Probability Rules for Compound Events

Law of Large Numbers: As an experiment is repeated many times, the empirical probability approaches the theoretical probability.
Range of Probabilities: All probabilities are between 0 and 1 inclusive (0 \le P(E) \le 1).
Complement of Events: The probability of "at least one" event occurring is 1 minus the probability that "none" of the events occur (P(\text{at least one}) = 1 - P(\text{none})).

Used when two or more events must both occur.
Independent Events: If the occurrence of event A does not affect the probability of event B, then P(A \text{ and } B) = P(A) \times P(B).
Dependent Events: If the occurrence of event A affects the probability of event B, then P(A \text{ and } B) = P(A) \times P(B|A).
Conditional Probability (P(B|A)): The probability of event B occurring given that event A has already occurred. When calculating from a chart, the denominator is the total of the "given" condition, not the overall total.

Used when at least one of two or more events occurs.
Mutually Exclusive Events: Events that cannot occur at the same time (P(A \text{ and } B) = 0). The probability is P(A \text{ or } B) = P(A) + P(B).
Not Mutually Exclusive Events: Events that can occur at the same time (they overlap). The probability is P(A \text{ or } B) = P(A) + P(B) - P(A \text{ and } B). The term - P(A \text{ and } B) subtracts the overlap to avoid double-counting.