A letter is known to have come either from KOLKATA or TATANAGAR. On t…

The correct answer is Option (2) → $\frac{3}{5}$

Let $E_1$: Letter from KOLKATA, $E_2$: Letter from TATANAGAR.

$P(E_1) = \frac12, \quad P(E_2) = \frac12$

Event $A$: The two consecutive letters "TA" are visible.

For KOLKATA:

Letters: K O L K A T A

Number of 2-letter sequences = $7-1=6$

"TA" occurs only once ⇒ $P(A|E_1) = \frac16$

For TATANAGAR:

Letters: T A T A N A G A R

Number of 2-letter sequences = $9-1=8$

"TA" occurs twice ⇒ $P(A|E_2) = \frac28 = \frac14$

Bayes' theorem:

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

$= \frac{\frac12 \cdot \frac14}{\frac12 \cdot \frac16 + \frac12 \cdot \frac14}$

$= \frac{\frac18}{\frac{1}{12} + \frac18}$

LCM of 12 and 8 is 24:

$= \frac{\frac{3}{24}}{\frac{2}{24} + \frac{3}{24}}$

$= \frac{3}{5}$