Answer:
a.
[tex]1 - P(A \cup B ) = 1 - 0.8 = 0.2[/tex]
b.
[tex]P(A \cap B) = P(A) + P(B) - P(A \cup B ) = 0.5 + 0.45 - 0.8 = 0.15[/tex]
c.
[tex]P(A \cap B^c ) = P(A) - P(A \cap B) = 0.5 - 0.15 = 0.35[/tex]
d.
They are not, the intersection is not 0.
Step-by-step explanation:
a.
The probability that the shopper has neither type of card is the probability of the complement of the union, therefore it would be
[tex]1 - P(A \cup B ) = 1 - 0.8 = 0.2[/tex]
b.
That's probability of the intersection of the events. For that we use the following formula
[tex]P(A \cup B ) = P(A) + P(B) - P(A \cap B)[/tex]
Therefore
[tex]P(A \cap B) = P(A) + P(B) - P(A \cup B ) = 0.5 + 0.45 - 0.8 = 0.15[/tex]
c.
That's the probability of A intersection the complement of B.
For that, first of all remember that
[tex]A = (A \cap B) \cup (A \cap B^c)[/tex]
Therefore
[tex]P(A) = P(A \cap B) + P(A \cap B^c)[/tex]
Therefore
[tex]P(A \cap B^c ) = P(A) - P(A \cap B) = 0.5 - 0.15 = 0.35[/tex]
d.
They are not, the intersection is not 0.
