question_answer

Let A and B be two events of the same sample space S of an experiment, than prove that \[0\le P\left( A/B \right)\le 1,\] \[B\ne \phi .\]

Answer:

By definition, \[P(A/B)=\frac{P(A\cap B)}{P(B)}\]       ?..(i) Also, \[A\cap B\subset B\]         \[[\because \,\,\,A\cap B)\,\,is\,\,a\,\,subset\,\,of\,\,B]\] \[\Rightarrow \]   \[P(A\cap B)\le P(B)\] \[\Rightarrow \]   \[\frac{P(A\cap B)}{P(B)}\le 1\]                                    ?.(ii) Again, \[P(A\cap B)\ge 0,\,and\,B\ne \phi \] \[\therefore \]      \[P(B)>0\] \[\therefore \]      \[\frac{P(A\cap B)}{P(B)}\ge 0\]                                       ?(iii) From Eqs. (ii) and (iii) we have \[0\le \frac{P(A\cap B)}{P(B)}\le 1\]         Hence, \[0\le P(A/B)\le 1\]                        [from Eq. (i)]                                                Hence proved.