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. $\frac{7}{11}$
• B. $\frac{2}{3}$
• C. $\frac{1}{5}$
• D. $\frac{1}{2}$

Answer 1

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

$\mathrm{Let}A\mathrm{be}\mathrm{the}\mathrm{event}\mathrm{that}\mathrm{the}\mathrm{letter}\mathrm{visible}\mathrm{is}\mathrm{TA}.\mathrm{Let}{E}_1\mathrm{and}{E}_2\mathrm{denote}\mathrm{that}\mathrm{the}\mathrm{letter}\mathrm{is}\mathrm{from}TATANAGAR\mathrm{and}CALCUTTA\mathrm{respectively}.$
$\mathrm{Then}P\left(E_1\right)=P\left(E_2\right)=\frac{1}{2}\mathrm{There}\mathrm{are}8\mathrm{pairs}\mathrm{of}\mathrm{consecutive}\mathrm{letters}\mathrm{in}TATANAGAR\mathrm{out}\mathrm{of}\mathrm{which}\mathrm{two}\mathrm{are}\mathrm{TA}.\therefore P\left(A|E_1\right)=\frac{2}{8}=\frac{1}{4}\mathrm{There}\mathrm{are}7\mathrm{pairs}\mathrm{of}\mathrm{consecutive}\mathrm{letters}\mathrm{in}CALCUTTA\mathrm{out}\mathrm{of}\mathrm{which}\mathrm{only}\mathrm{one}\mathrm{is}\mathrm{TA}.\therefore P\left(A|E_2\right)=\frac{1}{7}$
$\mathrm{We}\mathrm{need}\mathrm{to}\mathrm{find}P\left(E_1|A\right).\mathrm{By}\mathrm{Bayes}\text{'}\mathrm{theorem},\mathrm{we}\mathrm{have}P\left(E_1|A\right)=\frac{P\left(E_1\right)P\left(A|E_1\right)}{P\left(E_1\right)P\left(A|E_1\right)+P\left(E_2\right)P\left(A|E_2\right)}\Rightarrow P\left(E_1|A\right)=\frac{\frac{1}{2}\times \frac{1}{4}}{\frac{1}{2}\times \frac{1}{4}+\frac{1}{2}\times \frac{1}{7}}\Rightarrow P\left(E_1|A\right)=\frac{7}{11}$