Choosing Among the Probability Rules

Most probability problems only ask you to combine events with 'and,' 'or,' or 'at least one.' The hard part is never the arithmetic — it's matching the words to the right rule. 'Or' calls for the addition rule: P(A or B) = P(A) + P(B) - P(A and B), where you subtract the overlap unless the events are mutually exclusive, in which case there is no overlap to subtract. 'And' calls for the multiplication rule: P(A and B) = P(A) x P(B), but only if A and B are independent. If one event changes the chance of the other, you need the general version, P(A and B) = P(A) x P(B given A), which brings in conditional probability.

'At least one' problems have their own shortcut: the complement rule. Instead of adding up every way at least one event happens, find the probability that none happen, and subtract from 1. This works because 'at least one' and 'none' cover every possibility and never overlap. Deciding among these four tools starts with a question: does the problem involve overlap, independence, dependence, or the phrase 'at least one'? Read the scenario, identify the keyword, then check the condition — mutually exclusive, independent, or neither — before committing to a formula.

These rules are the same ones hiding inside every two-way table. A table's row and column totals give you the addition rule for 'or,' the cells give you 'and' directly, and dividing a cell by a row or column total gives conditional probability. Choosing among the rules here is really the same skill as reading a table: deciding what you're looking at before you calculate. This sorting skill is also what the binomial and normal models will lean on later, since both are built from repeated applications of these same four rules.