Probability
Probability is a number between 0 and 1 that measures how likely an event is to occur. For equally likely outcomes, it equals favorable outcomes divided by total possible outcomes — 0 means impossible, 1 means certain.
Probability quantifies uncertainty on a scale from 0 to 1. An event with probability 0 cannot happen, an event with probability 1 is certain, and everything else falls in between — a fair coin lands heads with probability 1/2. Probabilities can be written as fractions, decimals, or percents.
When all outcomes are equally likely, probability is a counting exercise: P(event) = favorable outcomes ÷ total outcomes. Rolling a standard die, P(even) = 3/6 = 1/2. From this definition come the workhorse rules. The complement rule says P(not A) = 1 − P(A), often the fastest route for "at least one" questions. The addition rule gives P(A or B) = P(A) + P(B) − P(A and B). The multiplication rule gives P(A and B) = P(A) × P(B) when A and B are independent; when they are not, the second factor becomes a conditional probability, as in drawing cards without replacement.
Probability underlies nearly all of statistics: probability distributions describe random variables, expected value averages outcomes weighted by their probabilities, and inference procedures interpret data through probability models.
Probability questions appear on a wide range of standardized tests. The GRE and ACT test single- and multi-event probability, complements, and independence, while AMC competition problems push further into clever counting, conditional probability, and geometric probability. Mastering the counting definition and the three basic rules covers most of what these exams ask.
Key takeaways
- Probability measures likelihood on a scale from 0 (impossible) to 1 (certain).
- With equally likely outcomes, P(event) = favorable outcomes ÷ total outcomes.
- The complement rule, P(not A) = 1 − P(A), is the fastest route for "at least one" problems.
- Multiply probabilities for independent events; use conditional probability when events affect each other.
- The GRE, ACT, and AMC all test probability through counting, complements, and independence.