Event: A subset of the sample space associated with a random experiment is called an event or a case.
e.g. In tossing a coin, getting either head or tail is an event.

Equally Likely Events: The given events are said to be equally likely if none of them is expected to occur in preference to the other.
e.g. In throwing an unbiased die, all the six faces are equally likely to come.

Mutually Exclusive Events: A set of events is said to be mutually exclusive, if the happening of one excludes the happening of the other, i.e. if A and B are mutually exclusive, then (A ∩ B) = Φ
e.g. In throwing a die, all the 6 faces numbered 1 to 6 are mutually exclusive, since if any one of these faces comes, then the possibility of others in the same trial is ruled out.

Exhaustive Events: A set of events is said to be exhaustive if the performance of the experiment always results in the occurrence of at least one of them.
If E1, E2, …, En are exhaustive events, then E1 ∪ E2 ∪……∪ En = S.
e.g. In throwing of two dice, the exhaustive number of cases is 62 = 36. Since any of the numbers 1 to 6 on the first die can be associated with any of the 6 numbers on the other die.

Three events E, F and G are said to be mutually independent, if
(i) P(E ∩ F) = P(E) . P(F)
(ii) P(F ∩ G) = P(F) . P(G)
(iii) P(E ∩ G) = P(E) . P(G)
(iv)P(E ∩ F ∩ G) = P(E) . P(F) . P(G)
If at least one of the above is not true for three given events, then we say that the events are not independent.
Note: Independent and mutually exclusive events do not have the same meaning.

Baye’s Theorem and Probability Distributions
Partition of Sample Space: A set of events E1, E2,…,En is said to represent a partition of the sample space S, if it satisfies the following conditions:
(i) Ei ∩ Ej = Φ; i ≠ j; i, j = 1, 2, …….. n
(ii) E1 ∪ E2 ∪ …… ∪ En = S
(iii) P(Ei) > 0, ∀ i = 1, 2,…, n

The theorem of Total Probability: Let events E1, E2, …, En form a partition of the sample space S of an experiment.If A is any event associated with sample space S, then