To determine the correct statement based on the given data, let's analyze the probabilities and relationships between a flower being pink and a flower being a rose.
First, compile the data from the table:
- Total number of flowers (denoted as [tex]\( T \)[/tex]) = 315
- Total number of pink flowers [tex]\( P_{\text{pink}} \)[/tex] = 60
- Total number of roses [tex]\( P_{\text{rose}} \)[/tex] = 105
- Total number of pink roses [tex]\( P_{\text{pink and rose}} \)[/tex] = 20
To check if a flower being pink and a flower being a rose are independent of each other, we need to compare the joint probability with the product of the individual probabilities.
1. Calculate the probability of a flower being pink:
[tex]\[
P(\text{Pink}) = \frac{P_{\text{pink}}}{T} = \frac{60}{315}
\][/tex]
2. Calculate the probability of a flower being a rose:
[tex]\[
P(\text{Rose}) = \frac{P_{\text{rose}}}{T} = \frac{105}{315}
\][/tex]
3. Calculate the joint probability of a flower being both pink and a rose:
[tex]\[
P(\text{Pink and Rose}) = \frac{P_{\text{pink and rose}}}{T} = \frac{20}{315}
\][/tex]
4. Calculate the product of the individual probabilities:
[tex]\[
P(\text{Pink}) \times P(\text{Rose}) = \left(\frac{60}{315}\right) \times \left(\frac{105}{315}\right)
\][/tex]
5. Compare the joint probability with the product of the individual probabilities:
[tex]\[
P(\text{Pink and Rose}) \stackrel{?}{=} P(\text{Pink}) \times P(\text{Rose})
\][/tex]
From the calculations (which we refer to the numerical result obtained earlier for confirmation), we know:
- The joint probability [tex]\( P(\text{Pink and Rose}) \)[/tex] is equal to [tex]\( \frac{20}{315} \)[/tex].
- The product of the individual probabilities [tex]\( P(\text{Pink}) \times P(\text{Rose}) \)[/tex] results in [tex]\( \frac{20}{315} \)[/tex].
Since
[tex]\[
P(\text{Pink and Rose}) = P(\text{Pink}) \times P(\text{Rose})
\][/tex]
we can conclude that a flower being pink and a flower being a rose are independent of each other.
Thus, the correct statement is:
A. A flower being pink and a flower being a rose are independent of each other.
