,"$undefined"]}]]}]}]}] 31:T10b9,

Experiment: An operation that can produce some well-defined outcomes.
Random Experiment: An experiment whose outcome cannot be predicted with certainty, but all possible outcomes are known.
Sample Space ($S$): The set of all possible outcomes of a random experiment.
Event ($E$): A subset of the sample space.
Elementary Event: An event having only one sample point.
Compound Event: An event having more than one sample point.
Impossible Event: An event that cannot occur ($\emptyset$).
Sure Event: An event that is certain to occur ($S$).

Classical (A Priori) Definition:
$P(E) = \frac{\text{Number of outcomes favorable to } E}{\text{Total number of possible outcomes}}$
For any event $E$, $0 \le P(E) \le 1$.
$P(\emptyset) = 0$ (Probability of an impossible event).
$P(S) = 1$ (Probability of a sure event).

Complement of an Event ($E'$ or $\bar{E}$): The event that $E$ does not occur.
$P(E') = 1 - P(E)$
$P(E) + P(E') = 1$

Two events $A$ and $B$ are mutually exclusive if they cannot occur simultaneously, i.e., $A \cap B = \emptyset$.
For mutually exclusive events: $P(A \cap B) = 0$.

A set of events $E_1, E_2, ..., E_n$ is exhaustive if their union is the sample space, i.e., $E_1 \cup E_2 \cup ... \cup E_n = S$.

If $E_1, E_2, ..., E_n$ are mutually exclusive and exhaustive events, then $P(E_1) + P(E_2) + ... + P(E_n) = 1$.

For any two events $A$ and $B$:
$P(A \cup B) = P(A) + P(B) - P(A \cap B)$
For three events $A, B, C$:
$P(A \cup B \cup C) = P(A) + P(B) + P(C) - P(A \cap B) - P(B \cap C) - P(C \cap A) + P(A \cap B \cap C)$
For mutually exclusive events $A$ and $B$:
$P(A \cup B) = P(A) + P(B)$

Two events $A$ and $B$ are independent if the occurrence of one does not affect the probability of the occurrence of the other.
Multiplication Rule for Independent Events:
$P(A \cap B) = P(A) \cdot P(B)$
If $A$ and $B$ are independent, then $A$ and $B'$, $A'$ and $B$, and $A'$ and $B'$ are also independent.

Union ($A \cup B$): Event $A$ or $B$ or both occur.
Intersection ($A \cap B$): Event $A$ and $B$ both occur.
Difference ($A - B$ or $A \cap B'$): Event $A$ occurs but $B$ does not occur.