Let's analyze the information provided and determine the relationship between "a flower being pink" and "a flower being a rose" using the data from the two-way table.
Here is the table again for reference:
[tex]\[
\begin{tabular}{|c|c|c|c|c|}
\hline
\text{Type of Flower/Color} & \text{Red} & \text{Pink} & \text{Yellow} & \text{Total} \\
\hline
\text{Rose} & 40 & 20 & 45 & 105 \\
\hline
\text{Hibiscus} & 80 & 40 & 90 & 210 \\
\hline
\text{Total} & 120 & 60 & 135 & 315 \\
\hline
\end{tabular}
\][/tex]
First, we need to calculate the expected frequency of pink roses if the events "being pink" and "being a rose" were independent. This is done by multiplying the total number of pink flowers by the total number of roses, and then dividing by the grand total:
[tex]\[
\text{Expected frequency of pink roses} = \frac{\text{(Total pink flowers)} \times \text{(Total roses)}}{\text{Grand total}} = \frac{60 \times 105}{315} = 20
\][/tex]
Next, compare the observed frequency of pink roses with the expected frequency:
- Observed frequency of pink roses: 20
- Expected frequency of pink roses (if independent): 20
Since the observed frequency (20) matches the expected frequency (20), the events "a flower being pink" and "a flower being a rose" are independent.
Therefore, the correct statement is:
A. A flower being pink and a flower being a rose are independent of each other.
