Answer :
To answer the question "Assuming someone has a flower garden, what is the probability they also have a vegetable garden?" we need to determine the conditional probability [tex]\( P(\text{Vegetable Garden} | \text{Flower Garden}) \)[/tex]. This is the probability that someone has a vegetable garden given that they already have a flower garden.
We will use the information provided in Table B, which details garden-type frequencies by rows. Here are the relevant components:
- [tex]\( P(\text{Flower Garden and Vegetable Garden}) \)[/tex]: This is the probability that someone has both a flower garden and a vegetable garden.
- [tex]\( P(\text{Flower Garden}) \)[/tex]: This is the probability that someone has a flower garden.
From Table B, we can clearly see:
- The entry in the Flower Garden row and Vegetable Garden column is [tex]\( 0.56 \)[/tex]. This represents [tex]\( P(\text{Flower Garden and Vegetable Garden}) = 0.56 \)[/tex].
- The total for the Flower Garden row is [tex]\( 1.0 \)[/tex]. This represents [tex]\( P(\text{Flower Garden}) = 1.0 \)[/tex].
The conditional probability formula is:
[tex]\[ P(\text{Vegetable Garden} | \text{Flower Garden}) = \frac{P(\text{Flower Garden and Vegetable Garden})}{P(\text{Flower Garden})} \][/tex]
Let's substitute the values:
[tex]\[ P(\text{Vegetable Garden} | \text{Flower Garden}) = \frac{0.56}{1.0} \][/tex]
Therefore,
[tex]\[ P(\text{Vegetable Garden} | \text{Flower Garden}) = 0.56 \][/tex]
So, assuming someone has a flower garden, the probability that they also have a vegetable garden is [tex]\( 0.56 \)[/tex] or 56%.
We will use the information provided in Table B, which details garden-type frequencies by rows. Here are the relevant components:
- [tex]\( P(\text{Flower Garden and Vegetable Garden}) \)[/tex]: This is the probability that someone has both a flower garden and a vegetable garden.
- [tex]\( P(\text{Flower Garden}) \)[/tex]: This is the probability that someone has a flower garden.
From Table B, we can clearly see:
- The entry in the Flower Garden row and Vegetable Garden column is [tex]\( 0.56 \)[/tex]. This represents [tex]\( P(\text{Flower Garden and Vegetable Garden}) = 0.56 \)[/tex].
- The total for the Flower Garden row is [tex]\( 1.0 \)[/tex]. This represents [tex]\( P(\text{Flower Garden}) = 1.0 \)[/tex].
The conditional probability formula is:
[tex]\[ P(\text{Vegetable Garden} | \text{Flower Garden}) = \frac{P(\text{Flower Garden and Vegetable Garden})}{P(\text{Flower Garden})} \][/tex]
Let's substitute the values:
[tex]\[ P(\text{Vegetable Garden} | \text{Flower Garden}) = \frac{0.56}{1.0} \][/tex]
Therefore,
[tex]\[ P(\text{Vegetable Garden} | \text{Flower Garden}) = 0.56 \][/tex]
So, assuming someone has a flower garden, the probability that they also have a vegetable garden is [tex]\( 0.56 \)[/tex] or 56%.
