Question

A letter is known to have come from either $\text{TATANAGAR}$ or $\text{CALCUTTA}$. On the envelope, just two consecutive letter $\text{TA}$ are visible. The probability that the letter has come from $\text{CALCUTTA}$ is

• A. $\frac{4}{11}$
• B. $\frac{1}{3}$
• C. $\frac{5}{12}$
• D. $\frac{1}{7}$

Answer 1

The correct option is A $\frac{4}{11}$

Let $T$ be the event that the letter has arrived from $TATANAGAR$ and $C$ be the event that the letter has arrived from $CALCUTTA$
$P\left(T\right)=\frac{1}{2},P\left(C\right)=\frac{1}{2}$

The probability that $TA$ comes from $TATANAGAR$ is,
$P\left(TA|T\right)=\frac{2}{8}$
The probability that $TA$ comes from $CALCUTTA$ is,
$P\left(TA|C\right)=\frac{1}{7}$

Using Bayes theorem :
$P\left(C|TA\right)=\frac{P\left(TA|C\right)P\left(C\right)}{P\left(TA|C\right)P\left(C\right)+P\left(TA|T\right)P\left(T\right)}$
$=\frac{\frac{1}{7}}{\frac{1}{7}+\frac{1}{4}}=\frac{4}{11}$