I need help explaning this practice problem: I am trying to study for a test. A

Question

I need help explaning this practice problem: I am trying to study for a test.

A university consists of four divisions, School of Business, School of Education, School of
Human Services and School of Liberal Arts and Sciences. The enrollment proportion of the
schools are about, 29.63%, 17.13%, 9.55% and 43.68%, respectively. Moreover, 87.91% of
business students are undergraduates, 77.48% of the education students are undergraduates,
52.88% of the human service students are undergraduates, and 96.21% of the LAS students
are undergraduates. Answer the following questions. Drawing a tree-diagram may help.

(a) If you select a student at random from this university, what is the probability that the selected student is an undergraduate student?

(b) If you select a student at random from this university, what is the probability that the selected student is a graduate student?

(c) If an undergraduate student is selected at random, what is the probability that the selected student belongs to School of Business? This was an extra credit problem.

Explanation / Answer

The simplest thing to do here is let the total no of admissions = x

The 2 tables below will give you a clear picture of the # of students in each school(Table1) and # of undergraduate and graduate students in table 2.

Table 2

There is no need to do the multiplication until we get the desired probabilty. We need to remember that

P= Total no. of favourable outcomes/Total no. of outcomes

(a) If you select a student at random from this university, what is the probability that the selected student is an undergraduate student?

Here the required probability is = Total no of Undergraduate students/ Total no of students

= ((29.63*x/100)*(87.94/100) +(17.13*x/100)*(77.48/100) + (9.55*x/100)*(52.88/100) + (43.68*x/100)*(96.21/100)) / x

= ((2605.66/10000) + (1327.33/10000) + (505/10000) + (4202.45/10000))*x / x

= 8640/10000

=0.864

(b) If you select a student at random from this university, what is the probability that the selected student is a graduate student?

Now here instead of adding all the students in the graduate column, and finding the probability all we need to do is P(picking a graduate student)= 1-P(Picking an undergraduate student)...WHY?? Because all the students are either Undergrads or graduates, therefore P(undergraduate) + P(Graduate) = 1

Therefore, P(picking a graduate student)= 1-0.864= 0.136

(c) If an undergraduate student is selected at random, what is the probability that the selected student belongs to School of Business?

Required Probability is = (Number of Undergraduate Students in school of Business) / No of undergraduate students

No of undergraduate students enrolling in school of business= (29.63*x/100)*(87.94/100)= 2605.66*x/10000

And total no of undergraduate students is =

= ((29.63*x/100)*(87.94/100) +(17.13*x/100)*(77.48/100) + (9.55*x/100)*(52.88/100) + (43.68*x/100)*(96.21/100)) / x

= ((2605.66/10000) + (1327.33/10000) + (505/10000) + (4202.45/10000))*x

= 8640*x/10000

Therefore required Probability

= (2605.66*x/10000) / (8640*x/10000)

= 2605.66/8640

= 0.301

School % # of students Business 29.63% 29.63*x/100 Education 17.13% 17.13*x/100 Human Service 9.55% 9.55*x/100 Arts and sciences 43.68% 43.68*x/100

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