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 $TATANAGAR$?

• A. $1/11$
• B. $7/9$
• C. $4/11$
• D. $7/11$

Answer 1

The correct option is D $7/11$

Let $E_1$ denote the event that the letter from $TATANAGAR$ and $E_2$ the event that the letter come from $CALCUTTA$.
Let $A$ denote the vent that the two consecutive alphabets visible on the envelop 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's theorem we have
$P\left(\frac{E_1}{A}\right)=\frac{P\left(E_1\right).P\left(\frac{A}{E_1}\right)}{P\left(E_1\right).P\left(\frac{A}{E_1}\right)+P\left(E_2\right).P\left(\frac{A}{E_2}\right)}=\frac{7}{11}$