Math 218
Final Exam Solved
Spring 2000
The solutions which we reproduce below are far more elaborate than what
was expected on the final. After all, not many students have laser printers and
Mathematica running on a laptop during the exam! Our purpose is to give as
full an explanation as possible, so
you
understand the solution.
Problem
1
.
(20 points) In a company of 200 employees, there are 32 employ
ees making at least $100,000 a year. There are 47 employees in the company
that have a graduate degree. There are 143 employees that do not have a
graduate degree and earn less than $100,000 per year.
(a) Find the probability that a randomly chosen employee makes less than
$100,000 and has a graduate degree.
(b) Find the probability that a randomly chosen employee makes at least
$100,000 or has a graduate degree.
(c) If an employee is selected at random, let
A
be the event that the em
ployee makes less than $100,000, and let
B
be the event that the em
ployee does not have a graduate degree. Find the value of
P
(
B

A
)
.
(d) Are the events
A
and
B
in part (c) independent? Justify your answer.
(e) Two employees are selected randomly to attend a lunch with the CEO
of the company. What is the probability they both have a graduate
degree?
Solution.
The easiest way to analyze this problem is to write down a table
summarizing the information which we know. We’ll use two columns, one
denoted
G
(for those with graduate degrees) and one denoted
G
(for those
without
graduate degrees; and two rows, one labeled
R
(for ‘Rich’, er, those
making $100,000 or more) and one labeled
R
(for those making less than
$100,000). We’ll mark the row sums to the right of the respective rows, and
the column sums just below the respective columns. In the lower right column
we enter 200, which should be both the row sum and the column sum for that
column.
Here’s the result:
G
G
Total
R
32
R
143
Total
47
200
The first thing we notice is that in order for the last row to sum to 200, the
column sum for
G
must be 153; and in order for the last column to sum to
200, the entry opposite
R
must be 168. So here’s what we know:
G
G
Total
R
32
R
143
168
Total
47
153
200
But this immediately allows us to fill in the
R
∩
G
entry (25
=
168
−
143)
and the
R
∩
G
entry (10
=
153
−
143):
G
G
Total
R
10
32
R
25
143
168
47
153
200
And finally we fill in the
R
∩
G
entry either from the row (22
=
32
−
10)
or
from the column (22
=
47
−
25):
G
G
Total
R
22
10
32
R
25
143
168
47
153
200
Now we’re ready to answer the questions.
(a) “Makes less than $100,000” is
R
; “has a graduate degree” is
G
; there
are 25 people in
R
∩
G
, as we can see from the final filledin table, so
the probability of randomly choosing one of these 25 from the 200 total
employees is 25
/
200, or 0
.
125.
(b) “Makes at least $100,000” is
R
; “has a graduate degree” is
G
. The
union of column
G
and row
R
contains 22
+
10
+
25
=
57 individuals.
Be careful that you don’t just add 47
+
32, the number in column
G
and
row
R
because this will doublecount the individuals who are in
R
∩
G
.