JEE Main & Advanced Mathematics Probability Classical Definition of Probability

If a random experiment results in n mutually exclusive, equally likely and exhaustive outcomes, out of which m are favourable to the occurrence of an event A, then the probability of occurrence of A is given by

\[P(A)=\frac{m}{n}=\frac{\text{Number of outcomes favourable to }A}{\text{Number of total outcomes}}\]

It is obvious that \[0\le m\le n\]. If an event A is certain to happen, then \[m=n,\] thus \[P(A)=1\].

If A is impossible to happen, then \[m=0\] and so \[P(A)=0\]. Hence we conclude that \[0\le P(A)\le 1\].

Further, if \[\bar{A}\] denotes negative of A i.e. event that A doesn’t happen, then for above cases m, n; we shall have

 \[P(\bar{A})=\frac{n-m}{n}=1-\frac{m}{n}=1-P(A)\] ,\[\therefore \]  \[P(A)+P(\bar{A})=1\].

Notations : For two events A and B,

(i) \[A'\] or \[\bar{A}\] or \[{{A}^{C}}\] stands for the non-occurrence or negation of A.

(ii) \[A\cup B\] stands for the occurrence of at least one of A and B.

(iii) \[A\cap B\] stands for the simultaneous occurrence of A and B.

(iv) \[A'\cap B'\] stands for the non-occurrence of both A and B.

(v) \[A\subseteq B\] stands for “the occurrence of A implies occurrence of B”.