A letter is known to come from 'TATANAGAR' or 'CALCUTTA' . In the envelope just two consecutive letters 'TA' are visible then the probability that the letter has came from 'TATANAGAR' is
Answer & explanation
Correct answer: option 4
The correct answer is Option (4) → \( \displaystyle\frac{7}{11} \)
Let \( E_1 \) be “the letter came from TATANAGAR” and \( E_2 \) “it came from CALCUTTA”, with \( P(E_1)=P(E_2)=\frac{1}{2} \). Let \( A \) be the event that “TA” is visible.
TATANAGAR has 9 letters, so 8 pairs of consecutive letters: TA, AT, TA, AN, NA, AG, GA, AR. Two of them are TA, so \( P(A|E_1)=\frac{2}{8}=\frac{1}{4} \).
CALCUTTA has 8 letters, so 7 pairs: CA, AL, LC, CU, UT, TT, TA. One of them is TA, so \( P(A|E_2)=\frac{1}{7} \).
By Bayes’ theorem,
\( \displaystyle P(E_1|A) \)
\( \displaystyle =\frac{\frac{1}{2}\cdot\frac{1}{4}}{\frac{1}{2}\cdot\frac{1}{4}+\frac{1}{2}\cdot\frac{1}{7}} \)
\( \displaystyle =\frac{\frac{1}{4}}{\frac{11}{28}} \)
\( \displaystyle =\frac{7}{11} \)
NCERT Exemplar:
Probability
