Question

In a school in Florida, 60% of the students stay in the school's dormitory and 40% stay with their families. The school records show that 30% of the students living in the dormitory and 20% of the students living with their families obtain As on exams. If a student chosen at random from the school receives As, the probability that the student lives in the school dormitory is .....

A. 1/26
B. 1/18
C. 9/26
D. 9/13

Answer 1

We have events:


`D` - a student stays in the school's dormitory

`D'` - a student stays with family

`A` - a student receives As

and probabilities:


`P(D)=0.6\\\\P(D')=0.4\\\\P(A|D)=0.3\\\\P(A|D')=0.2`

We want to calculate the probability that the student lives in the school dormitory given he receives As so it will be
`P(D|A)`. From the Bayes' theorem we know that:


`P(D|A)=(P(A|D)P(D))/(P(A))`

The only thing we don't know is
`P(A)`, but we can calculate it using the law of total probability. There will be:


`P(A)=P(A|D)P(D)+P(A|D')P(D')=0.3\cdot0.6+0.2\cdot0.4=\\\\=0.18+0.08=\boxed{0.26}`

So our probability:


`P(D|A)=(P(A|D)P(D))/(P(A))=(0.3\cdot0.6)/(0.26)=(0.18)/(0.26)=\boxed{(9)/(13)}`

Answer D.