Question

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

• A. $\frac{6}{11}$
• B. $\frac{5}{11}$
• C. $\frac{4}{11}$
• D. $\frac{7}{11}$

Answer 1

The correct option is D $\frac{7}{11}$

Let $E_1$ be the event that letter is from $TATANAGAR$ and $E_2$ be the event that letter is from $CALCUTTA$.
Let $E$ be the event that on letter two consecutive letter $TA$ are visible.
$P\left(E_1\right)=\frac{1}{2},P\left(E_2\right)=\frac{1}{2}$
$TATANAGAR$ have following set of consecutive letters {$TA,AT,TA,AN,NA,AG,GA,AR$}
$CALCUTTA$ have the following set of consecutive letters {$CA,AL,LC,CU,UT,TT,TA$}
$P\left(\frac{E}{E_1}\right)=\frac{2}{8},P\left(\frac{E}{E_2}\right)=\frac{1}{7}$
Using Bayes Theorem,
$P\left(E_1/E\right)=\left(\frac{P\left(E_1\right).P\left(E/E_1\right)}{P\left(E_1\right).P\left(E/E_1\right)+P\left(E_2\right).P\left(E/E_2\right)}\right)$
$P\left(\frac{E_1}{E}\right)=\left[\frac{\frac{1}{2}\times \frac{2}{8}}{\frac{1}{2}\times \frac{2}{8}+\frac{1}{2}\times \frac{1}{7}}\right]=\frac{14}{22}=\frac{7}{11}$

beLet E1 be the event that letter is from TATA NAGAR and E2 be the event that letter is from CALCUTTA.Read more on Sarthaks.com - https://www.sarthaks.com/143105/a-letter-is-known-to-have-come-either-from-tata-nagar-or-from-calcutta