A wooden cube whose faces are painted red is cut by 2 horizontal, 2 vertical planes and 2 planes in another direction so as to get 27 cubes of same dimensions. One cube is selected at random and rolled. Let  $E_i$ be the event that the selected cube contains exactly i faces painted red; $[i=0,1,2,3]$ .Let  ‘A’ be the event that the face with red colour to turn up. Then 

Number of faces painted red                                    Number of cubes
               $E_0:0$                                                                         1
              $E_1:1$                                                                          6
              $E_2:2$                                                                        12
              $E_3:3$                                                                          8
Using Baye’s theorem
 $p\left(\frac{E_1}{A}\right)=\left[\frac{(\frac{6}{27}\times \frac{1}{6})}{(\frac{1}{27}\times 0)+(\frac{6}{27}\times \frac{1}{6})+(\frac{12}{27}\times \frac{2}{6})+(\frac{8}{27}\times \frac{3}{6})}\right]=\frac{1}{9}$
 $P\left(\frac{{\overline{E}}_1}{A}\right)=1-P\left(\frac{E_1}{A}\right)=1-\frac{1}{9}=\frac{8}{9}$
$\overline{A}$ =Event of getting blank face to turn up 
Using Baye’s theorem,
 $p\left(\frac{E_1}{\overline{\underset{\_}{A}}}\right)=\left[\frac{(\frac{6}{27}\times \frac{5}{6})}{(\frac{1}{27}\times \frac{6}{6}+\frac{6}{27}\times \frac{5}{6}+\frac{12}{27}\times \frac{4}{6})+(\frac{8}{27}\times \frac{3}{6})}\right]=\frac{(6\times 5)}{6\times (1+5+8+4)}=\frac{5}{18}$