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Question
40% students of a college reside in hostel and the remaining reside outside. At the end of the year, 50% of the hostelers got A grade while from outside students, only 30% got A grade in the examination. At the end of the year, a student of the college was chosen at random and was found to have gotten A grade. What is the probability that the selected student was a hosteler ?
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Solution
Let E1 and E2 be the events that the student is a hosteller or an outside student, respectively and A be the event that the chosen student gets A grade.
∴P(E1)=40%=40/100=0.4
P(E2)=(100−40)%=60%=60/100=0.6
P(A|E1)=P(Student getting A grade is a hosteller)=50%=0.5
P(A|E2)=P(Student getting A grade is an outside student)=30%=0.3
The probability that a randomly chosen student is a hosteller, given that he got A grade, is given by P(E1|A).
Using Bayes’ theorem, we get
`P(E_1|A)=(P(E1)⋅P(A|E_1))/(P(E_1)⋅P(A|E_1)+P(E_2)⋅P(A|E_2))`
`=(0.4xx0.5)/(0.4xx0.5+0.6xx0.3)`
`=0.20/0.38 `
`=10/19`
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