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Question

Of the students in a college, it is known that 60% reside in hostel and 40% are day scholars (not residing in hostel). Previous year results report that 30% of all students who reside in hostel attain A grade and 20% of day scholars attain A grade in their annual examination. At the end of the year, one student is chosen at random from the college and he has an A grade, what is the probability that the student is a hosteller?

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Solution

Let E1 and E2 be the events that the student is a hostler and a day scholar respectively and A be the event that the chosen student gets grade A.

P(E1)=60% =60100=0.6

P(E2)=40% =40100=0.4

P(A|E1)=P(student getting an A grade is a hostler)=30% =0.3
P(A|E2)=P(student getting an A grade is a day scholar)=20% =0.2

The probability that a randomly chosen student is a hostler, given that he has an A grade, is given by P(E1|A).
By using Baye's theorem, we obtain

P(E1|A)=P(E1)P(A|E1)P(E1)P(A|E1)+P(E2)P(A|E2)

=0.6×0.30.6×0.3+0.4×0.2

=0.180.26

=1826

=913=0.69

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