Question

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

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

Answer 1

The correct option is 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 envelops are $TA$.
We have $P\left(E_1\right)=1/2,P\left(E_2\right)=1/2,P\left(A|E_1\right)=2/8,P\left(A|E_2\right)=1/7$.
Therefore, by Baye's theorem we have
$P\left(E_2|A\right)=\frac{P\left(E_2\right)P\left(A|E_2\right)}{P\left(E_1\right)P\left(A|E_1\right)+P\left(E_2\right)P\left(A|E_2\right)}=\frac{4}{11}$.