Name Probability
A letter is known to have come either from 'TATA NAGAR' or from 'CALCUTTA' . On the envelope, just two consecutive letters TA are visible. What is the probability that the letter came from 'TATA NAGAR' ? The answer will be in the form of , where and are coprime positive integers, find .
The answer is 18.
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Let A be the event that the letter is from TATA NAGAR and B be the event that letter is from CALCUTTA .
Also, let E be the event that on the letter, two consecutive letters TA are visible.
P(A) = 2 1 ; P(B) = 2 1
And P(E/A) = 8 2 and P(E/B) = 7 1
[ If the letter is TATA NAGAR, we see that the events of two consecutive letters visible are { TA, AT, TA, AN, NA, AG, GA, AR }. So P(E/A) = 8 2 and same in case of CALCUTTA, so P(E/B) = 7 1 ]
Therefore, P(A/E) = P ( A ) . P ( E / A ) + P ( B ) . P ( E / B ) P ( A ) . P ( E / A )
= [ ( 1 / 2 ) ∗ ( 2 / 8 ) ] + [ ( 1 / 2 ) ∗ ( 1 / 7 ) ] ( 1 / 2 ) ∗ ( 2 / 8 )
= ( 1 / 8 ) + ( 1 / 1 4 ) ( 1 / 8 )
= ( 1 1 / 5 6 ) ( 1 / 8 )
= 1 1 7
p = 7 , q = 11; p+q = 11 + 7 = 18