Probability of opposite, mutually exclusive, and overlapping events

Learning Objectives Define and identify opposite events. Calculate the probability of an opposite event. Distinguish between mutually exclusive and overlapping events. Apply the addition rule for mutually exclusive events. Apply the addition rule for overlapping events. Solve real-world problems involving opposite, mutually exclusive, and overlapping probabilities. Have you ever wondered about the chances of something NOT happening, or the chances of one thing OR another happening? 🤔 Let's explore the math behind these everyday questions! In this lesson, you'll learn how to calculate the likelihood of different types of events: those that are opposites, those that can't happen at the same time, and those that can. Understanding these concepts helps you make...

TermDefinitionExample ProbabilityA measure of how likely an event is to occur, expressed as a number between 0 and 1 (or 0% and 100%).The probability of flipping a coin and getting heads is 1/2 or 50%. EventA specific outcome or a set of outcomes in an experiment.When rolling a standard die, 'rolling an even number' is an event (outcomes: 2, 4, 6). Opposite Event (Complement)The event that an event A does NOT occur. It includes all outcomes in the sample space that are not in A.If event A is 'rolling a 3' on a die, the opposite event (A') is 'not rolling a 3' (outcomes: 1, 2, 4, 5, 6). Mutually Exclusive EventsTwo events that cannot happen at the same time. If one occurs, the other cannot.When rolling a die, 'rolling an even number' and 'r...

Probability of an Event $P(A) = \frac{\text{Number of favorable outcomes}}{\text{Total number of outcomes}}$ This is the fundamental rule for calculating the probability of any single event A. Probability of an Opposite Event (Complement Rule) $P(A') = 1 - P(A)$ or $P(\text{not A}) = 1 - P(A)$ To find the probability that an event A does NOT happen, subtract the probability of A happening from 1 (or 100%). Probability of Mutually Exclusive Events (Addition Rule) $P(A \text{ or } B) = P(A) + P(B)$ If two events A and B cannot happen at the same time, the probability that A OR B occurs is the sum of their individual probabilities. Probability of Overlapping Events (General Addition Rule) $P(A \text{ or } B) = P(A) + P(B) - P(A \text{ and } B)$ If two eve...