Question

A letter is known to have come either from $TATANAGAR$ or $CALCUTTA$. On the envelope just two consecutive letters $TA$ are visible. What is the probability that the letter came from $CALCUTTA$?

• A. $\frac{4}{11}$
• B. $\frac{7}{11}$
• C. $\frac{2}{11}$
• D. None of these
  

Answer 1

Answer: (A)

$\frac{4}{11}$

Let $E_1$ denote the event that the letter came from $TATANAGAR$ and $E_2$ the event that the letter came from $CALCUTTA.$
Let $A$ denote the event that the two consecutive alphabets visible on the envelope are $TA.$
We have $P\left(E_1\right)=\frac{1}{2},P\left(E_2\right)=\frac{1}{2},P\left(\frac{A}{E_1}\right)=\frac{2}{8},P\left(\frac{A}{E_2}\right)=\frac{1}{7},$
Therefore, by Bayes' theorem we have
$\left(\frac{E_2}{A}\right)=\frac{P\left(E_2\right)P\left(\frac{A}{E_2}\right)}{P\left(E_1\right)P\left(\frac{A}{E_1}\right)+P\left(E_2\right)P\left(\frac{A}{E_1}\right)}=\frac{4}{11}$