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Question
A letter is known to have come either from TATA NAGAR or from CALCUTTA. On the envelope, just two consecutive letter TA are visible. What is the probability that the letter came from TATA NAGAR.
Solution
Let E1: The event that the letter comes from TATA NAGAR
And E2: The event that the letter comes from CALCUTTA
Also E3: The event that on the letter, two consecutive letters TA are visible
∴ P(E1) = `1/2` and P(E2) = `1/2`
And `"P"("E"_3/"E"_1) = 2/8` and `"P"("E"_3/"E"_2) = 1/7` ......[∵ For TATA NAGAR, the two consecutive letters visible are TA, AT, TA, AN, NA, AG, GA, AR]
∴ `"P"("E"_3/"E"_1) = 2/8`
And [For CALCUTTA, the two consecutive letters visible are CA, AL, LC, CU, UT, TT and TA]
So, `"P"("E"_3/"E"_2) = 1/7`
Now using Bayes’ Theorem, we have
`"P"("E"_1/"E"_3) = ("P"("E"_1)*"P"("E"_3/"E"_1))/("P"("E"_1)*"P"("E"_3/"E"_1) + "P"("E"_2) * "P"("E"_3/"E"_2))`
= `(1/2*2/8)/(1/2*2/8 + 1/2*1/7)`
= `(1/8)/(1/8 + 1/14)`
= `(1/8)/((7 + 4)/56)`
= `7/11`
Hence, the required probability is `7/11`.
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