Probability and Complementary Events

Objective: Count favorable and total equally likely outcomes.

Prerequisite skills: fractions and event counts. For uniform selection from a finite set, probability equals favorable count divided by total count. The event and sample space must use the same unit of outcome, such as one token or one person. Common error: using an unrelated count as the denominator. Restrictions and interpretation: Assume uniform selection only when the prompt states or implies each listed item is equally likely.

Objective: Use one minus a probability for the event not occurring.

An event and its complement partition the sample space, so their probabilities add to one. The complement of at least one is none, not exactly one. Carefully negate the event wording. Common error: subtracting from 100 without converting between percent and probability. Restrictions and interpretation: Assume uniform selection only when the prompt states or implies each listed item is equally likely.

Objective: Count either event without double-counting their intersection.

For inclusive either-or, add the event probabilities and subtract their overlap once. If events are mutually exclusive, the overlap is zero. Do not assume disjointness merely because the labels differ. Common error: adding overlapping groups without subtracting the overlap. Restrictions and interpretation: Assume uniform selection only when the prompt states or implies each listed item is equally likely. Desmos and method choice: Use Desmos for arithmetic or a graph/table check when that reduces work. Enter parentheses explicitly and verify the requested quantity and units. Use the algebraic steps shown here for an exact result; a rounded graph display is not proof of exact equality.