A letter is known to have arrived by post either from KANPUR or...

Let $E_1$ be the event that the letter came from KANPUR and $E_2$ be the event that the letter came from ANANTPUR.

Since the letter is equally likely to come from either city, $P(E_1) = P(E_2) = \dfrac{1}{2}$.

Let $A$ be the event that the two consecutive visible letters are AN.

In the word KANPUR, there are 6 letters, so there are 5 pairs of consecutive letters: KA, AN, NP, PU, UR. The pair AN appears exactly once. Thus, $P(A|E_1) = \dfrac{1}{5}$.

In the word ANANTPUR, there are 8 letters, so there are 7 pairs of consecutive letters: AN, NA, AN, NT, TP, PU, UR. The pair AN appears exactly twice. Thus, $P(A|E_2) = \dfrac{2}{7}$.

By Bayes' theorem, the probability that the letter came from ANANTPUR given that AN is visible is:

$P(E_2|A) = \dfrac{P(E_2) P(A|E_2)}{P(E_1) P(A|E_1) + P(E_2) P(A|E_2)}$

$P(E_2|A) = \dfrac{\dfrac{1}{2} \times \dfrac{2}{7}}{\dfrac{1}{2} \times \dfrac{1}{5} + \dfrac{1}{2} \times \dfrac{2}{7}}$

$P(E_2|A) = \dfrac{\dfrac{2}{7}}{\dfrac{1}{5} + \dfrac{2}{7}}$

$P(E_2|A) = \dfrac{\dfrac{2}{7}}{\dfrac{17}{35}} = \dfrac{2}{7} \times \dfrac{35}{17} = \dfrac{10}{17}$

Answer: $\dfrac{10}{17}$