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.

Answer 1

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`.