AP Statistics Unit 4 Probability Rules: How to Compute and Approximate Probabilities

Simulation

A method to estimate a probability by mimicking a chance process many times and using long-run relative frequency as an approximation of the true probability.

Theoretical probability

An exact probability found using probability rules and counting methods (not by running trials).

Empirical probability

A probability estimated from data or repeated random trials (such as a simulation).

Probability model (for simulation)

A clear specification of how randomness enters a situation: possible outcomes, their probabilities, and assumptions like equal likelihood or independence.

Randomization method

The tool used to generate chance outcomes in a simulation (e.g., random digits table, random number generator, coins/dice when appropriate).

Trial (in simulation)

One complete repetition of the chance process with a stated stopping point and a recorded outcome.

Success condition

The precise rule stating what outcome counts as a “success” in each simulation trial.

Relative frequency estimate (p-hat)

The simulation probability estimate computed as the number of successes divided by the number of trials: p̂ = s/n.

Law of Large Numbers

As the number of trials increases, the sample proportion of successes tends to get closer to the true probability (though small samples can vary a lot).

Random digits mapping

A rule that assigns random numbers (e.g., 00–99) to outcomes so that the number ranges match the intended probabilities.

Independence assumption (in simulation)

The assumption that one trial’s outcome (or one step’s outcome) does not affect another, unless the context indicates dependence.

Event

A set of outcomes from a chance process (something that may or may not occur).

Union (A ∪ B)

The event that A occurs or B occurs (or both); in probability, “or” is usually inclusive.

Intersection (A ∩ B)

The event that both A and B occur (“and”).

Mutually exclusive (disjoint) events

Events that cannot happen at the same time; characterized by P(A ∩ B) = 0.

General Addition Rule

A rule for “or” probabilities: P(A ∪ B) = P(A) + P(B) − P(A ∩ B).

Addition Rule for mutually exclusive events

If A and B are disjoint, then P(A ∪ B) = P(A) + P(B) because P(A ∩ B) = 0.

Overlap (double-counting) in addition

When adding P(A)+P(B), outcomes in both A and B are counted twice, so you subtract P(A ∩ B) once.

Venn diagram (probability)

A visual reasoning tool showing unions, intersections, and overlaps of events; useful for applying the addition rule.

Conditional probability

The probability of an event given that another event has occurred; it restricts the sample space to the “given” event.

Conditional probability formula

For P(B) > 0, P(A | B) = P(A ∩ B) / P(B).

Two-way table

A table organizing counts or proportions for two categorical variables; often the clearest way to compute conditional probabilities.

Independent events

Events where knowing one occurs does not change the probability of the other; e.g., P(A | B) = P(A) (when P(B) > 0).

Independence test (multiplication form)

A and B are independent if P(A ∩ B) = P(A)P(B).

General Multiplication Rule

A rule for “and” probabilities: P(A ∩ B) = P(B)P(A | B) (or equivalently P(A)P(B | A)).