Question

The registrar needs to assign section numbers to a group of 10 sophomores, 20 juniors, and 30 seniors. There are three sections with available seats, say section A, B, C. Each section has room for up to 60 students. We will assume that the registrar randomly and independently assigns the students to the sections.

a) What is the probability that all ten sophomores will be assigned to the same section?

b) What is the probability that exactly 15 students will be assigned to section B?

c) Given that exactly 15 students are assigned to section B, what is the probability that these 15 students are all juniors?

d) Given that exactly 15 students are assigned to section B, what is the probability that these 15 students consist of 5 sophomores, 5 juniors, and 5 seniors?

Answer 1

Solution:

There are three sections with available seats, say section A, B, C. Each section has room for up to 60 students.

So, Total Number of ways =

(a) We have to find the probability that all ten sophomores will be assigned to the same section ,

Number of ways of assign one out of three sections is 3.

Now, there are 50 students ( juniors and seniors)

probability that all ten sophomores will be assigned to the same section

  probability that all ten sophomores will be assigned to the same section

(b) Number of ways to select 15 students from total number of students is

Now , 60-15 = 45 students

Number of ways of assign two sections to 45 students is given as

Now,

the probability that exactly 15 students will be assigned to section B is given by,

the probability that exactly 15 students will be assigned to section B

(c) Number of selecting 15 juniors out of 20 is

the probability that these 15 students are all juniors

(d) Now,

Number of ways to select 5 sophomores, 5 juniors, and 5 seniors is

the probability that these 15 students consist of 5 sophomores, 5 juniors, and 5 seniors is,

All parts are solved.

Please rating the answer. Thank you!