Probability
What you must understand
Probability measures how likely an event is, on a scale from 0 to 1. For equally likely outcomes, P(event) = (number of favourable outcomes) / (total outcomes). The complement rule says P(not A) = 1 − P(A). For two events, the addition rule is P(A ∪ B) = P(A) + P(B) − P(A ∩ B); if the events are mutually exclusive, the intersection is zero. Independent events satisfy P(A and B) = P(A) × P(B).
Know the difference between experimental (empirical) probability, based on observed data, and theoretical probability, based on counting equally likely outcomes. Understand sample spaces, tree diagrams, and two-way tables, and be able to read conditional probability from a table by restricting to the given row or column.
Traps to watch for
- Adding probabilities of overlapping events without subtracting the intersection.
- Assuming events are independent when they are not (e.g., drawing without replacement).
- Using the wrong denominator in a conditional-probability question.
- Confusing "at least one" with "exactly one"; use the complement for "at least one."
- Reporting a probability greater than 1 or negative, a sign of an arithmetic error.
Rule of thumb
For "at least one," compute 1 − P(none).
Worked example: Two fair dice are rolled. P(sum is 7) = 6/36 = 1/6, since the favourable pairs are (1,6),(2,5),(3,4),(4,3),(5,2),(6,1). P(sum is not 7) = 1 − 1/6 = 5/6.