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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 hostler?
Solution
E1, E2 and A represent the following:
E1 = students living in hostels,
E2 = Students not residing in hostels
and A = students who get A-grades
Now P(E1) = `60/100 = 3/5`, P(E2) = `40/100 = 2/5`
P(A|E1) = `30/100 = 3/10`, P(A|E2) = `20/100 = 2/10`
By Bayes' theorem
P(E1|A) = `(P(A|E_1) P(E_1))/(P(A|E_1)P(E_1) + P(A|E_2)P(E_2))`
`= (3/10 xx 3/5)/((3/10 xx 3/5) + (2/10 xx 2/5))`
`= 9/ (9 + 4)`
`= 9/13`
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