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Please solve qtn 14 a with explanation..dont give any links plz..

Question 14. (a) A letter is known to have come either from TATANAGAR or CALCUTTA. On the envelope, just two consecutive letter TA are visible. What is the probability that the letter has come from (i) Calcutta ? (ii) Tatanagar ?
 
                     (b) A coin is tossed 5 times. What is the probability of getting at least 3 heads ? 
 

Let E1 denote the event that the letter came from TATANAGAR and E2 the event that the letter came from CALCUTTA.

Let A denote the event that the two consecutive alphabets visible on the envelope are TA.

Since, the letters have come either from Calcutta or Tatanagar, Therefore,

P (E1) = ½ and P (E2) = ½

 

If E1 has occurred, then it means that the letter came from TATANAGAR. In the word TATANAGAR there are 8 consecutive alphabets. i.e., {TA, AT, TA, AN, NA, AG, GA and AR} and TA occurs two time.

Therefore, P (A|E1) = 2/8=1/4

 

If E2 has occurred, then it means that the letter came from Calcutta. In the word CALCUTTA there are 7 consecutive alphabets. i.e., {CA, AL, LC, CU, UT, TT and TA} and TA occurs only one time.

Therefore, P (A|E2) = 1/7

 

P (A|E1) = 2/8, P (A|E2) = 1/7. Therefore, by Bayes theorem we have,

Hence, probability that the letter has come from CALCUTTA is 4/11.

And,

Hence, probability that the letter has come from TATA NAGAR is 7/11.
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Let E1 denote the event that the letter came from TATANAGAR and E2 the event that the letter came from CALCUTTA. Let A denote the event that the two consecutive alphabets visible on the envelope are TA. Since, the letters have come either from Calcutta or Tatanagar, Therefore, P (E1) = ½ and P (E2) = ½ If E1 has occurred, then it means that the letter came from TATANAGAR. In the word TATANAGAR there are 8 consecutive alphabets. i.e., {TA, AT, TA, AN, NA, AG, GA and AR} and TA occurs two time. Therefore, P (A|E1) = 2/8=1/4 If E2 has occurred, then it means that the letter came from Calcutta. In the word CALCUTTA there are 7 consecutive alphabets. i.e., {CA, AL, LC, CU, UT, TT and TA} and TA occurs only one time. Therefore, P (A|E2) = 1/7 P (A|E1) = 2/8, P (A|E2) = 1/7. Therefore, by Bayes theorem we have, Hence, probability that the letter has come from CALCUTTA is 4/11. And, Hence, probability that the letter has come from TATA NAGAR is 7/11.
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