A Probability Course for the Actuaries A Preparation for Exam P/1

28 BASIC OPERATIONS ON SETS
Problems
Problem 2.1
Let A and B be any two sets. Use Venn diagram to show that B = (A ∩
B) ∪ (A c ∩ B) and A ∪ B = A ∪ (A c ∩ B).
Problem 2.2
Show that if A ⊆ B then B = A ∪ (A c ∩ B). Thus, B can be written as the
union of two disjoint sets.
Problem 2.3
A survey of a group’s viewing habits over the last year revealed the following
information
(i)
(ii)
(iii)
(iv)
(v)
(vi)
(vii)
28% watched gymnastics
29% watched baseball
19% watched soccer
14% watched gymnastics and baseball
12% watched baseball and soccer
10% watched gymnastics and soccer
8% watched all three sports.
Represent the statement “the group that watched none of the three sports
during the last year” using operations on sets.
Problem 2.4
An urn contains 10 balls: 4 red and 6 blue. A second urn contains 16 red
balls and an unknown number of blue balls. A single ball is drawn from each
urn. For i = 1, 2, let R i denote the event that a red ball is drawn from urn
i and B i the event that a blue ball is drawn from urn i. Show that the sets
R 1 ∩ R 2 and B 1 ∩ B 2 are disjoint.
Problem 2.5 ‡
An auto insurance has 10,000 policyholders. Each policyholder is classified
as
(i)
(ii)
(iii)
young or old;
male or female;
married or single.