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Anuj s.
Let E1 denote the event that the letter came from TATANAGAR and E2 denote the event that the letter came from CALCUTTA. Let A be the event that the two consecutive alphabets visible on the envelope are TA. Since the letters have come from either Calcutta or Tatanagar. Therefore, P(E1)=21,P(E2)=21 If E1 has occurred then the letter has come from TATANAGAR. In the word TATANAGAR there are 8 consecutive alphabets i.e.,{TA,AT,TA,AN,NA,AG,GA,AR} and TA occurs two times. ∴P(A/E1)=82=41 If E2 has occurred then the letter has come from CALCUTTA. In the word CALCUTTA there are 7 consecutive alphabets i.e.,{CA,AL,LC,CU,UT,TT,TA} and TA occurs once. ∴P(A/E2)=71 The probability that the letter has come from TATANAGAR is P(E1/A) By Bayes theorem, P(E1/A)=P(E1)P(A/E1)+P(E2)P(A/E2)P(E1)P(A/E1) =21×41+21+7121×41 =81+14181 ∴P(E1/A)=117 Hence the probability that the letter has come from TATANAGAR is 117.