Math / Data & Statistics

QuestionReal estate ads suggest that $51 \%$ of homes for sale have garages, $36 \%$ have swimming pools, and $33 \%$ have both features. What percentage of homes for sale have a) a pool or a garage?
Answer = $\square$ \% b) neither a pool nor a garage?
Answer = $\square$ \% c) a pool but no garage?
Answer = $\square$ \%
Note: Enter your answers as percentages, not decimals.

Studdy Solution

STEP 1

1. Let $P(G)$ be the probability that a home has a garage.
2. Let $P(P)$ be the probability that a home has a swimming pool.
3. Let $P(G \cap P)$ be the probability that a home has both a garage and a swimming pool.
4. Use the principle of inclusion-exclusion to find probabilities for unions and complements.

STEP 2

1. Calculate the percentage of homes with either a pool or a garage.
2. Calculate the percentage of homes with neither a pool nor a garage.
3. Calculate the percentage of homes with a pool but no garage.

STEP 3

Use the principle of inclusion-exclusion to find the percentage of homes with either a pool or a garage:
$P(G \cup P) = P(G) + P(P) - P(G \cap P)$
Substitute the given values:
$P(G \cup P) = 51\% + 36\% - 33\%$
Calculate:
$P(G \cup P) = 54\%$

STEP 4

To find the percentage of homes with neither a pool nor a garage, use the complement rule:
$P(\text{neither}) = 1 - P(G \cup P)$
Substitute the calculated value:
$P(\text{neither}) = 100\% - 54\%$
Calculate:
$P(\text{neither}) = 46\%$

STEP 5

To find the percentage of homes with a pool but no garage, calculate:
$P(P \cap \neg G) = P(P) - P(G \cap P)$
Substitute the given values:
$P(P \cap \neg G) = 36\% - 33\%$
Calculate:
$P(P \cap \neg G) = 3\%$
The answers are: a) $54\%$ b) $46\%$ c) $3\%$

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