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A letter is known to have come either from TATANAGAR or from CALCUTTA. On the envelope just two consecutive letters TA are visible. What is the probability that the letters came from TATANAGAR ?
- a)7/11
- b)4/11
- c)24/29
- d)1/11
Correct answer is option 'A'. Can you explain this answer?
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Verified Answer
A letter is known to have come either from TATANAGAR or from CALCUTTA....
Ans.
Option (a)
Let E1 denote the event that the letter came from TATA NAGAR 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 Tata nagar, Therefore,
P (E1) = � and P (E2) = �
If E1 has occurred, then it means that the letter came from TATA NAGAR. In the word TATA NAGAR 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.
Most Upvoted Answer
A letter is known to have come either from TATANAGAR or from CALCUTTA....
Given:
A letter is known to have come either from TATANAGAR or from CALCUTTA.
On the envelope just two consecutive letters TA are visible.
To find:
The probability that the letters came from TATANAGAR.
Solution:
Let's assume that the letter came from TATANAGAR (event A).
So, the probability of the letter coming from CALCUTTA (event B) is 1 - P(A) = P(B).
Let's count the number of possibilities:
- TATANAGAR has 9 letters.
- CALCUTTA has 8 letters.
- We can see two consecutive letters TA on the envelope.
For TATANAGAR, there are 2 TAs, TA_ _ _ _ _ _ _, where _ denotes any letter.
For CALCUTTA, there are 3 TAs, TA_ _ _ _ _ _ _ , _TA_ _ _ _ _ _ , and _ _TA_ _ _ _ _ , where _ denotes any letter.
The total number of possibilities is 2 + 3 = 5.
The probability of the letter coming from TATANAGAR (event A) is the number of favorable outcomes (2) divided by the total number of outcomes (5):
P(A) = 2/5
The probability of the letter coming from CALCUTTA (event B) is:
P(B) = 1 - P(A) = 3/5
Therefore, the probability that the letters came from TATANAGAR is:
P(A|TA) = P(A and TA) / P(TA)
where P(TA) is the probability of observing TA on the envelope.
The probability of observing TA on the envelope is:
P(TA) = P(A and TA) + P(B and TA)
where P(A and TA) is the probability that the letter came from TATANAGAR and TA is observed on the envelope, and P(B and TA) is the probability that the letter came from CALCUTTA and TA is observed on the envelope.
For TATANAGAR, there are 2 TAs and 7 other letters, so the probability of observing TA on the envelope is:
P(A and TA) = (2/9) * (1/8) = 1/36
For CALCUTTA, there are 3 TAs and 6 other letters for each possibility, so the probability of observing TA on the envelope is:
P(B and TA) = [(3/8) * (1/7)] + [(3/7) * (1/8)] = 9/224
Therefore, the probability of observing TA on the envelope is:
P(TA) = (1/36) + (9/224) = 47/1008
Now, we can calculate the probability that the letters came from TATANAGAR given that TA is observed on the envelope:
P(A|TA) = (2/5) * (1/36) / (47/1008) = 7/11
Therefore, the probability that the letters came from TATANAGAR given that TA is observed on the envelope is 7/11 or option (a).
A letter is known to have come either from TATANAGAR or from CALCUTTA.
On the envelope just two consecutive letters TA are visible.
To find:
The probability that the letters came from TATANAGAR.
Solution:
Let's assume that the letter came from TATANAGAR (event A).
So, the probability of the letter coming from CALCUTTA (event B) is 1 - P(A) = P(B).
Let's count the number of possibilities:
- TATANAGAR has 9 letters.
- CALCUTTA has 8 letters.
- We can see two consecutive letters TA on the envelope.
For TATANAGAR, there are 2 TAs, TA_ _ _ _ _ _ _, where _ denotes any letter.
For CALCUTTA, there are 3 TAs, TA_ _ _ _ _ _ _ , _TA_ _ _ _ _ _ , and _ _TA_ _ _ _ _ , where _ denotes any letter.
The total number of possibilities is 2 + 3 = 5.
The probability of the letter coming from TATANAGAR (event A) is the number of favorable outcomes (2) divided by the total number of outcomes (5):
P(A) = 2/5
The probability of the letter coming from CALCUTTA (event B) is:
P(B) = 1 - P(A) = 3/5
Therefore, the probability that the letters came from TATANAGAR is:
P(A|TA) = P(A and TA) / P(TA)
where P(TA) is the probability of observing TA on the envelope.
The probability of observing TA on the envelope is:
P(TA) = P(A and TA) + P(B and TA)
where P(A and TA) is the probability that the letter came from TATANAGAR and TA is observed on the envelope, and P(B and TA) is the probability that the letter came from CALCUTTA and TA is observed on the envelope.
For TATANAGAR, there are 2 TAs and 7 other letters, so the probability of observing TA on the envelope is:
P(A and TA) = (2/9) * (1/8) = 1/36
For CALCUTTA, there are 3 TAs and 6 other letters for each possibility, so the probability of observing TA on the envelope is:
P(B and TA) = [(3/8) * (1/7)] + [(3/7) * (1/8)] = 9/224
Therefore, the probability of observing TA on the envelope is:
P(TA) = (1/36) + (9/224) = 47/1008
Now, we can calculate the probability that the letters came from TATANAGAR given that TA is observed on the envelope:
P(A|TA) = (2/5) * (1/36) / (47/1008) = 7/11
Therefore, the probability that the letters came from TATANAGAR given that TA is observed on the envelope is 7/11 or option (a).
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Community Answer
A letter is known to have come either from TATANAGAR or from CALCUTTA....
Ans.
Option (a)
Let E1 denote the event that the letter came from TATA NAGAR 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 Tata nagar, Therefore,
P (E1) = � and P (E2) = �
If E1 has occurred, then it means that the letter came from TATA NAGAR. In the word TATA NAGAR 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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A letter is known to have come either from TATANAGAR or from CALCUTTA. On the envelope just two consecutive letters TA are visible. What is the probability that the letters came from TATANAGAR ?a)7/11b)4/11c)24/29d)1/11Correct answer is option 'A'. Can you explain this answer?
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