Key Points

A random experiment is an action with an uncertain outcome. The set of all possible outcomes is called the sample space, denoted by S.

An event is any subset of a sample space S. For example, getting an even number when rolling a die is the event $E = \{2, 4, 6\}$.

An impossible event is represented by the empty set $\phi$, and its probability is $P(\phi) = 0$. A sure event is the entire sample space S, and its probability is $P(S) = 1$.

A simple or elementary event is an event with only one sample point. A compound event is an event with more than one sample point.

The event 'A or B' represents the union of two events, $A \cup B$. It occurs if A occurs, or B occurs, or both occur.

The event 'A and B' represents the intersection of two events, $A \cap B$. It occurs only if both A and B occur simultaneously.

The event 'not A', or the complementary event of A, is denoted by $A'$. It represents all outcomes in the sample space S that are not in A, so $A' = S - A$.

Two events A and B are mutually exclusive if they cannot happen at the same time. Mathematically, their intersection is empty: $A \cap B = \phi$.

Events $E_1, E_2, ..., E_n$ are exhaustive if their union forms the entire sample space: $E_1 \cup E_2 \cup ... \cup E_n = S$. This means at least one of these events must occur.

Probability P is a function that satisfies three axioms: (i) For any event E, $P(E) \ge 0$. (ii) $P(S) = 1$. (iii) If E and F are mutually exclusive events, $P(E \cup F) = P(E) + P(F)$.

For an experiment with a finite sample space S where all outcomes are equally likely, the probability of an event A is given by $P(A) = \frac{n(A)}{n(S)}$, where $n(A)$ is the number of outcomes favorable to A.

The probability of the event 'not A' is calculated using the probability of event A. The formula is $P(A') = 1 - P(A)$.

For any two events A and B, the probability of 'A or B' is given by the formula $P(A \cup B) = P(A) + P(B) - P(A \cap B)$.

If events A and B are mutually exclusive, then $P(A \cap B) = 0$. The addition rule simplifies to $P(A \cup B) = P(A) + P(B)$.