A letter is known to have come either from LONDON or CLIFTON; on the postmark only the two consecutive letters ON are legible. The probability that it came from LONDON is

We define the following events:
${\mathrm{A}}_1:$Selecting a pair of consecutive letter from the word LONDON
${\mathrm{A}}_2:$Selecting a pair of consecutive letters from the word CLIFTON
E : Selecting a pair of letters 'ON'.
Then $\mathrm{P}\left({\mathrm{A}}_1\cap \mathrm{E}\right)=\frac{2}{5}$; as there are 5 pairs of consecutive letters out of which 2 are ON. 
$\mathrm{P}\left({\mathrm{A}}_2\cap \mathrm{E}\right)=\frac{1}{6}$; as there are 6 pairs of consecutive letters of which one is ON.
$\therefore$the required probability is $\mathrm{P}\left(\frac{{\mathrm{A}}_1}{\mathrm{E}}\right)=\frac{\mathrm{P}\left({\mathrm{A}}_1\cap \mathrm{E}\right)}{\mathrm{P}\left({\mathrm{A}}_1\cap \mathrm{E}\right)+\mathrm{P}\left({\mathrm{A}}_2\cap \mathrm{E}\right)}=\frac{\frac{2}{5}}{\frac{2}{5}+\frac{1}{6}}=\frac{12}{17}$