Explanation
Step 1: Define the Events
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Let E1: The letter came from TATANAGAR.
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Let E2: The letter came from CALCUTTA.
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Let A: The visible consecutive letters are TA.
Since it is equally likely for the letter to come from either city:
Step 2: Find the number of consecutive pairs in each word
To find the probability of seeing "TA", we count all possible pairs of consecutive letters.
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For TATANAGAR:
The pairs are: TA, AT, TA, AN, NA, AG, GA, AR.
Total pairs = 8
Number of "TA" pairs = 2 (at positions 1-2 and 3-4)
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For CALCUTTA:
The pairs are: CA, AL, LC, CU, UT, TT, TA.
Total pairs = 7
Number of "TA" pairs = 1 (at positions 7-8)
Step 3: Apply Bayes' Theorem
We need to find the probability that the letter came from CALCUTTA given that "TA" is visible (P(E2∣A)):
P(E2∣A)=P(E1)⋅P(A∣E1)+P(E2)⋅P(A∣E2)P(E2)⋅P(A∣E2)
Substitute the values:
P(E2∣A)=(21⋅41)+(21⋅71)21⋅71
Cancel the common factor of 21:
Step 4: Simplify the fraction
Calculate the denominator: 41+71=287+4=2811
Conclusion:
The probability that the letter has come from CALCUTTA is 4/11.
Correct Option: (a)
