(A) Let $E_{1}$ be the event that the student is a hostler and $E_{2}$ be the event that the student is a day scholar. Let $A$ be the event that the c

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(A) Let $E_{1}$ be the event that the student is a hostler and $E_{2}$ be the event that the student is a day scholar. Let $A$ be the event that the chosen student gets an $A$ grade.Given probabilities:$P(E_{1}) = 60 \% = 0.6$$P(E_{2}) = 40 \% = 0.4$$P(A | E_{1}) = 30 \% = 0.3$$P(A | E_{2}) = 20 \% = 0.2$We need to find $P(E_{1} | A)$. By Bayes' theorem:$P(E_{1} | A) = \frac{P(E_{1}) \cdot P(A | E_{1})}{P(E_{1}) \cdot P(A | E_{1}) + P(E_{2}) \cdot P(A | E_{2})}$Substituting the values:$P(E_{1} | A) = \frac{0.6 \times 0.3}{(0.6 \times 0.3) + (0.4 \times 0.2)}$$P(E_{1} | A) = \frac{0.18}{0.18 + 0.08}$$P(E_{1} | A) = \frac{0.18}{0.26}$$P(E_{1} | A) = \frac{18}{26} = \frac{9}{13}$

Class 12 Mathematics Probability Baye's theorem

Medium

Of the students in a college,it is known that $60 \%$ reside in a hostel and $40 \%$ are day scholars (not residing in a hostel). Previous year results report that $30 \%$ of all students who reside in a 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 they have an $A$ grade. What is the probability that the student is a hostler?

A
$\frac{9}{13}$
B
$\frac{8}{13}$
C
$\frac{7}{13}$
D
$\frac{6}{13}$

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Probability

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Baye's theorem

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