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You have a (universal) set U of 80 members and the subsets
- H of 32 members (own at least one hybrid car)
- E of 14 members (own at least one electric car) and
- HE of 4 members (own at least one hybrid and at least one electric car).
Notice that the subset HE is the intersection of subsets H and E.
(a) If a member of the club is surveyed, what is the probability that he or she owns only hybrid cars?
The number of those who own only hybrid car is H - HE = 32 - 4 = 28.
The probability under the question is = = 0.35 = 35%.
(b) If a member of the club is surveyed, what is the probability that he or she owns no alternative fuel vehicles?
The number of those who own at least one of hybrid or at least one electric car is
H + E - HE = 32 + 14 - 4 = 42. (*)
The rest of the 80 members of the club, i.e. 80 - 42 = 38 members, do not own any AFV car.
So the probability under this question is = = 0.475 = 47.5%.
Solved.
In this solution, only the formula (*) requires explanation, and the explanation is very simple.
In any finite set, the number of elemenst of the UNION of any two its subsets is equal to the sum of the numbers of elements
in each subset minus the number of elements in the intersection (since when we add the cardinalities of the subsets,
we count the intersection members twice).
See the lesson
- Counting elements in sub-sets of a given finite set
in this site.