Q.3  Of the students in a college, it is known that  60^{o}/_{o}  reside in hostel and  40^{o}/_{o}  are day scholars (not residing in hostel). Previous year results report that  30^{o}/_{o}  of all students who reside in hostel attainA grade and  20^{o}/_{o}  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 hostlier?

Answers (1)

H :   reside in hostel

D :  day scholars

A : students who attain grade A 

P(H)=\frac{60}{100}=\frac{6}{10}=\frac{3}{5}

P(D)=\frac{40}{100}=\frac{4}{10}=\frac{2}{5}

P(A|H)=\frac{30}{100}=\frac{3}{10}

P(A|D)=\frac{20}{100}=\frac{2}{10}= \frac{1}{5}

By Bayes theorem :

P(H|A)=\frac{P(H).P(A|H)}{P(H).P(A|H)+P(D).P(A|D)}

                 P(H|A)=\frac{\frac{3}{5}\times \frac{3}{10}}{\frac{3}{5}\times \frac{3}{10}+\frac{2}{5}\times \frac{1}{5}}

                  P(H|A)=\frac{\frac{9}{50}}{\frac{9}{50}+\frac{2}{25}}

                   P(H|A)=\frac{\frac{9}{50}}{\frac{13}{50}}

                 P(H|A)=\frac{9}{13}

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