**Random Experiment**

An experiment whose outcomes cannot be predicted or determined in advance is called a random experiment.

**Outcome**

A possible result of a random experiment is called its outcome.

**Sample Space**

A sample space is the set of all possible outcomes of an experiment.

**Events**

An event is a subset of a sample space associated with a random experiment.

**Types of Events**

**Impossible and sure events:** The empty set Φ and the sample space S describes events. Intact Φ is called the impossible event and S i.e. whole sample space is called sure event.

**Simple or elementary event:** Each outcome of a random experiment is called an elementary event.

**Compound events:** If an event has more than one outcome is called compound events.

**Complementary events:** Given an event A, the complement of A is the event consisting of all sample space outcomes that do not correspond to the occurrence of A.

**Mutually Exclusive Events**

Two events A and B of a sample space S are mutually exclusive if the occurrence of any one of them excludes the occurrence of the other event. Hence, the two events A and B cannot occur simultaneously and thus P(A ∩ B) = 0.

**Exhaustive Events**

If E_{1}, E_{2},…….., E_{n} are n events of a sample space S and if E_{1} ∪ E_{2} ∪ E_{3} ∪………. ∪ E_{n} = S, then E_{1}, E_{2},……… E_{3} are called exhaustive events.

**Mutually Exclusive and Exhaustive Events**

If E_{1}, E_{2},…… E_{n} are n events of a sample space S and if

E_{i} ∩ E_{j} = Φ for every i ≠ j i.e. E_{i} and E_{j} are pairwise disjoint and E_{1} ∪ E_{2} ∪ E_{3} ∪………. ∪ E_{n} = S, then the events

E_{1}, E_{2},………, E_{n} are called mutually exclusive and exhaustive events.

**Probability Function**

Let S = (w_{1}, w_{2},…… w_{n}) be the sample space associated with a random experiment. Then, a function p which assigns every event A ⊂ S to a unique non-negative real number P(A) is called the probability function.

It follows the axioms hold

- 0 ≤ P(w
_{i}) ≤ 1 for each W_{i}∈ S - P(S) = 1 i.e. P(w
_{1}) + P(w_{2}) + P(w_{3}) + … + P(w_{n}) = 1 - P(A) = ΣP(w
_{i}) for any event A containing elementary event w_{i}.

**Probability of an Event**

If there are n elementary events associated with a random experiment and m of them are favorable to an event A, then the probability of occurrence of A is defined as

The odd in favour of occurrence of the event A are defined by m : (n – m).

The odd against the occurrence of A are defined by n – m : m.

The probability of non-occurrence of A is given by P() = 1 – P(A).

**Addition Rule of Probabilities**

If A and B are two events associated with a random experiment, then

P(A ∪ B) = P(A) + P(B) – P(A ∩ B)

Similarly, for three events A, B, and C, we have

P(A ∪ B ∪ C) = P(A) + P(B) + P(C) – P(A ∩ B) – P(A ∩ C) – P(B ∩ C) + P(A ∩ B ∩ C)

Note: If A andB are mutually exclusive events, then

P(A ∪ B) = P(A) + P(B)