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.

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.