JEE Main & Advanced Mathematics Probability Definitions of Various Terms

(1) Sample space : The set of all possible outcomes of a trial (random experiment) is called its sample space. It is generally denoted by S and each outcome of the trial is said to be a sample point.

(2) Event : An event is a subset of a sample space.

(i) Simple event : An event containing only a single sample point is called an elementary or simple event.

(ii) Compound events : Events obtained by combining together two or more elementary events are known as the compound events or decomposable events.

(iii) Equally likely events : Events are equally likely if there is no reason for an event to occur in preference to any other event.

(iv) Mutually exclusive or disjoint events : Events are said to be mutually exclusive or disjoint or incompatible if the occurrence of any one of them prevents the occurrence of all the others.

 (v) Mutually non-exclusive events : The events which are not mutually exclusive are known as compatible events or mutually non exclusive events.

(vi) Independent events : Events are said to be independent if the happening (or non-happening) of one event is not affected by the happening (or non-happening) of others.

(vii) Dependent events : Two or more events are said to be dependent if the happening of one event affects (partially or totally) other event.

(3) Exhaustive number of cases : The total number of possible outcomes of a random experiment in a trial is known as the exhaustive number of cases.

(4) Favourable number of cases : The number of cases favourable to an event in a trial is the total number of elementary events such that the occurrence of any one of them ensures the happening of the event.

(5) Mutually exclusive and exhaustive system of events :  Let S be the sample space associated with a random experiment. Let \[{{A}_{1}},{{A}_{2}},\text{ }\ldots ..{{A}_{n}}\] be subsets of S such that

(i) \[{{A}_{i}}\cap {{A}_{j}}=\varphi \] for \[i\ne \,j\] and (ii) \[{{A}_{1}}\cup {{A}_{2}}\cup ....\cup {{A}_{n}}=S\]

Then the collection of events \[{{A}_{1}},\,{{A}_{2}},\,.....,\,{{A}_{n}}\] is said to form a mutually exclusive and exhaustive system of events.

If \[{{E}_{1}},\,{{E}_{2}},\,.....,\,{{E}_{n}}\] are elementary events associated with a random experiment, then

(i) \[{{E}_{i}}\cap {{E}_{j}}=\varphi \] for \[i\ne \,j\] and (ii) \[{{E}_{1}}\cup {{E}_{2}}\cup ....\cup {{E}_{n}}=S\] 

So, the collection of elementary events associated with a random experiment always form a system of mutually exclusive and exhaustive system of events.

In this system, \[P({{A}_{1}}\cup {{A}_{2}}.......\cup {{A}_{n}})\]

\[=P({{A}_{1}})+P({{A}_{2}})+.....+P({{A}_{n}})=1\].