Math 362K Probability Fall 2007 Instructor: Geir Helleloid
Daily Homework 5: Solutions
1. (Chapter 2, Problem 13) A certain town of population size 100,000 has 3 newspapers:
I, II, and III. The proportions of townspeople who read these papers are as follows:
I: 10 percent I and II: 8 percent I and II and III: 1 percent
II: 30 percent I and III: 2 percent
III: 5 percent II and III: 4 percent
(The list tells us, for instance, that 8000 people read newspapers I and II.)
(a) Find the number of people who read only one newspaper.
Solution. Let the sample space Sconsist of the 100,000 townspeople. Let E1be
the event consisting of the people who read I, let E2be the event consisting of
the people who read II, and let E3be the event consisting of the people who read
III. Using the given table of percentages, we find
|E1|= 10,000 |E1∩E2|= 8,000 |E1∩E2∩E3|= 1,000
|E2|= 30,000 |E1∩E3|= 2,000
|E3|= 5,000 |E2∩E3|= 4,000
The number of people who read only one newspaper is the total number of towns-
people minus the answers to (b) and (d). So the answer is 20,000 .
(b) How many people read at least two newspapers?
Solution. The people who read at least two newspapers are (E1∩E2)∪(E1∩
E3)∪(E2∩E3). By Inclusion-Exclusion,
|(E1∩E2)∪(E1∩E3)∪(E2∩E3)|
=|E1∩E2|+|E1∩E3|+|E2∩E3|−|(E1∩E2)∩(E1∩E3)|
−|(E1∩E2)∩(E2∩E3)|−|(E1∩E3) ∩(E2∩E3)|
+|(E1∩E2)∩(E1∩E3)∩(E2∩E3)|
= 8,000 + 2,000 + 4,000 −1,000 −1,000 −1,000 + 1,000
= 12,000
(c) If I and III are morning papers and II is an evening paper, how many people read
at least one morning paper plus an evening paper?
