AP Statistics: Probability Rules and Foundations Summary

The sample space represents the set of all possible outcomes for a random phenomenon.
When outcomes are formed by sequential independent choices with $n_1, n_2, \dots, n_k$ options, the size of the sample space is the product of the number of options at each step ( $n_1 \times n_2 \times \dots \times n_k$ ).

Two events $A$ and $B$ are mutually exclusive (disjoint) if they cannot occur at the same time, meaning their joint probability is zero: $P(A \cap B) = 0$ .
If $P(A \cap B) \neq 0$ , the events are not mutually exclusive.
When categories represent non-overlapping partitions of a population where every individual selects exactly one category, the categories are mutually exclusive.

General Addition Rule: For any two events $A$ and $B$ , the probability that at least one occurs is given by $P(A \cup B) = P(A) + P(B) - P(A \cap B)$ .
Solving for Joint Probability: Rearranging the addition rule gives the joint probability of two events: $P(A \cap B) = P(A) + P(B) - P(A \cup B)$ .
Complement Rule: The probability that an event $A$ does not occur is $P(A^c) = 1 - P(A)$ .
For two mutually exclusive events $A$ and $B$ , the probability that neither occurs is $1 - (P(A) + P(B))$ .

Conditional Probability Formula: The probability of event $A$ given that event $B$ has occurred is defined as $P(A|B) = \frac{P(A \cap B)}{P(B)}$ , provided $P(B) > 0$ .
In two-way contingency tables, conditional probability is calculated by restricting the sample space to the row or column total corresponding to the given condition.

General Multiplication Rule: The joint probability of two events occurring together is $P(A \cap B) = P(A) \cdot P(B|A)$ .
Dependent Events (Sampling Without Replacement): When sampling without replacement, conditional probabilities must be updated after each selection to reflect the changing composition of the remaining pool.
Independent Events: Two events $A$ and $B$ are independent if the occurrence of one does not affect the probability of the other: $P(A|B) = P(A)$ and $P(A \cap B) = P(A) \cdot P(B)$ .
Independence of Complements: If events $A$ and $B$ are independent, their complements $A^c$ and $B^c$ are also independent: $P(A^c \cap B^c) = P(A^c) \cdot P(B^c)$ .