Chapter 4: Probability

Beginning with basic definitions, probability is presented as a numerical measure between zero and one that quantifies likelihood, where outcomes combine to form events, and equally likely selections are called equiprobable. The concept of exhaustive events and complements establishes that an event and its negation partition all possibilities, with their probabilities summing to unity. Expectation is then introduced as the predicted frequency of occurrence across repeated trials, calculated by multiplying the number of trials by the event probability. The chapter progresses to analyzing relationships between events, distinguishing mutually exclusive events that cannot co-occur from those sharing common outcomes. For mutually exclusive events, the addition law simplifies probability calculations by summing individual probabilities, while the general addition law accounts for overlapping events by subtracting the intersection probability to avoid double-counting. Venn diagrams are presented as visual representations for organizing event spaces and understanding unions, intersections, and complements. The multiplication law addresses independent events where one event's occurrence does not influence another, allowing joint probability calculation through direct multiplication. Tree diagrams and outcome tables provide systematic methods for computing combined probabilities and identifying relative frequencies. Conditional probability introduces the dependence of probabilities on given information, using specialized notation and interpretation. The distinction between independent and dependent events frames the difference between sampling with replacement versus without replacement. Finally, the general multiplication law extends probability reasoning to dependent scenarios, incorporating conditional probability to calculate joint outcomes when the second event's probability depends on the first. Throughout, formulas and diagrammatic tools enable students to translate real-world probability problems into computational procedures.