AP - Statistics:: Probability, Random Variables, and Probability Distributions

The LAW OF LARGE NUMBERS states that:

- **A)** Short-run sequences of random outcomes always average out immediately
- **B)** Probabilities change depending on the number of trials conducted
- **C)** As the number of trials of a random process increases, the proportion of times a particular outcome occurs approaches its true probability
- **D)** Each individual random outcome becomes more predictable as more trials occur

SIMULATION is useful for estimating probabilities when:

- **A)** The exact probability is already known with certainty
- **B)** Simulations always give the exact same result as a mathematical calculation
- **C)** The exact theoretical probability is difficult to calculate directly, so repeatedly modeling the random process (e.g., with random digits or technology) can approximate the true probability
- **D)** Simulation can only be used for categorical variables

The PROBABILITY of an event is formally defined as a number between 0 and 1 that represents:

- **A)** The exact outcome of a single trial
- **B)** The number of possible outcomes in the sample space
- **C)** The long-run proportion of times that event would occur if the random process were repeated infinitely many times
- **D)** A number that can range from 0 to 100

Two events are MUTUALLY EXCLUSIVE (disjoint) if:

- **A)** They are always independent of each other
- **B)** They always occur together
- **C)** Their probabilities always sum to exactly 1
- **D)** They cannot both occur at the same time (their intersection is empty)

For two mutually exclusive events \(A\) and \(B\), the ADDITION RULE states:

- **A)** \(P(A \text{ or } B) = P(A) \times P(B)\)
- **B)** \(P(A \text{ or } B) = P(A) + P(B)\)
- **C)** \(P(A \text{ or } B) = 1\) always
- **D)** \(P(A \text{ or } B) = P(A) - P(B)\)

For events that are NOT necessarily mutually exclusive, the GENERAL ADDITION RULE states:

- **A)** \(P(A \text{ or } B) = P(A \text{ and } B)\)
- **B)** \(P(A \text{ or } B) = P(A) + P(B) - P(A \text{ and } B)\)
- **C)** \(P(A \text{ or } B) = P(A) + P(B)\)
- **D)** \(P(A \text{ or } B) = P(A) \times P(B)\)

CONDITIONAL PROBABILITY, written \(P(A|B)\), represents:

- **A)** The probability that event \(A\) occurs, GIVEN that event \(B\) has already occurred (or is known to be true)
- **B)** The probability that neither \(A\) nor \(B\) occurs
- **C)** The probability that \(A\) causes \(B\)
- **D)** The probability that both \(A\) and \(B\) occur

The formula for conditional probability is:

- **A)** \(P(A|B) = P(A) \cdot P(B)\)
- **B)** \(P(A|B) = \dfrac{P(A \text{ and } B)}{P(B)}\), provided \(P(B)>0\)
- **C)** \(P(A|B) = P(B) - P(A)\)
- **D)** \(P(A|B) = P(A) + P(B)\)