A particular state has elected both a governor and a senator. Let A be the event that a randomly selected voter has a favorable view of a certain party’s senatorial candidate, and let B be the corresponding event for that party’s gubernatorial candidate.
Suppose that P(A′) = .44, P(B′) = .57, and P(A ⋃ B) = .68 (these figures are suggested by the 2010 general election in California).
(a) What is the probability that a randomly selected voter has a favorable view of both candidates?
(b) What is the probability that a randomly selected voter has a favorable view of exactly one of these candidates?
(c) What is the probability that a randomly selected voter has an unfavorable view of at least one of these candidates?

Given that A be the event that a randomly selected voter has a favorable view of a certain party’s senatorial candidate, and let B be the corresponding event for that party’s gubernatorial candidate.

Suppose that

P(A′) = .44, P(B′) = .57, and P(A ⋃ B) = .68

From the above we can find out

P(A) = [tex]1-0.44 = 0.56[/tex]

P(B) = [tex]1-0.57 = 0.43[/tex]

P(AUB) = 0.68 =

[tex]0.56+0.43-P(A\bigcap B)\\P(A\bigcap B)=0.30[/tex]

a) the probability that a randomly selected voter has a favorable view of both candidates=P(AB) = 0.30

b) the probability that a randomly selected voter has a favorable view of exactly one of these candidates

= P(A)-P(AB)+P(B)-P(AB)

[tex]=0.99-0.30-0.30\\=0.39[/tex]

c) the probability that a randomly selected voter has an unfavorable view of at least one of these candidates

=P(A'UB') = P(AB)'

=[tex]1-0.30\\=0.70[/tex]