A group of 8 industrious students at Random University is ta

A group of 8 industrious students at Random University is taking the Probability Circus class this semester. To accomodate the stringent study habits of these students, the president of the university has set aside k (k < 8) individual study rooms in the campus library for the exclusive use by any of the 8 students. The study pattern of the students is as follows: each student, independently of any other, will study at home with probability 0.4 or at the university with probability 0.6; if a student chooses to study at the university, he/she will study in an individual room with probability 0.7 or elsewhere in the campus with probability 0.3. Determine the number of rooms k that must be set aside so that the probability that a student can find an empty room is greater than 0.9.

Solution

Probability of study at home : P(H) = 0.4

Probability of study at university : P(U) = 0.6

Probability of study in indivisual room given inside university : P(IR/U) = 0.7

Probability of study elsewhere in campus given inside university : P(C/U) = 0.3

Therefore ,

P(IR intersection U) = 0.7*0.6 = 0.42

P(C intersection U) = 0.3 * 0.6 = 0.18

Number of rooms =>

0.9 = 0.42 *k

or, k = 0.9/0.42 = 2.14 ~ 3 Rooms