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9 September, 21:50

Real estate ads suggest that 56 % of homes for sale have garages, 27 % have swimming pools, and 16 % have both features. a) What is the probability that a home for sale has a garage, but not a pool? b) If a home for sale has a garage, what's the probability that it has a pool, too? c) Are having a garage and having a pool independent events? Explain. d) Are having a garage and having a pool mutually exclusive? Explain.

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  1. 9 September, 22:00
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    a) 0.4

    b) 0.28

    c) The events are not independent

    d) The events are not mutually exclusive

    Step-by-step explanation:

    Hi!

    Lets call:

    Gar = {homes with garage}

    Pool = {homes with pool},

    A = {homes with pool and garage} = Gar ∩ Pool

    The data we are given is:

    P (Gar) = 0.56

    P (Pool) = 0.27

    P (A) = 0.16

    a) B = {homes with garage but not pool}. This set B is the set Gar without set A: B = Gar / A

    P (B) = P (Gar) - P (A) = 0.4

    b) This is a conditional probability:

    P (Pool | Gar) = P (Gar ∩ Pool) / P (Gar) = 0.16/0.56 = 0.28

    c) To be independent events, it must be, by definition:

    P (Gar ∩ Pool) = P (Gar) * P (Pool)

    0.16 ≠ 0.56*0.27 = 0.15

    Then, the events Gar and Pool are not independent

    d) Gar and Pool are not mutually exclusive, because there are houses with both pool and garage. We know that because P (A) is not zero.
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