Question

forty-five percent of the students in a dorm are business majors and fifty-five percent are non-business majors. business majors are twice as likely to do their studying at the library as non-business majors are. half of the business majors study at the library. if a randomly slected student from the dorm studies at the library, what is the probability the student is a business major

Answer 1

Solution :

Defining the following events as :

B : Being a Business major

α : Studying at the library

∴ Given that :

$P(B) = \frac{45}{100}$

         = 0.45

Again, P [ Studying at the library | Being a Business major ] = 2 P [ Studying at the library | Being a non business major ]

$P[ \alpha | B] = 2 P[\alpha | B^C]$   .......(1)

Again,

$P[\text{Studying at the library } | \text{ Being a business major}] = \frac{1}{2} = 0.50$

$P(\alpha | B) = 0.50$

From (1), we get

$P(\alpha | B^C) = \frac{1}{2} . P(\alpha | B)$

              $=\frac{1}{2} \times 0.50$

              = 0.25

Therefore, we need,

= P[  The students is a Business major | The student studies at the library ]

$=P(B | \alpha)$

By Bayes theorem

$=\frac{P(B). P(\alpha | B)}{P(B).P(\alpha | B) + P(B^C). P(\alpha | B^C)}$

$=\frac{0.45 \times 0.50}{0.45 \times 0.50 + 0.55 \times 0.25}$

= 0.6207