To determine the correct statement regarding the relationship between being a pink flower and being a rose, we will analyze the probabilities from the given data.
The table shows:
- There are 315 total flowers.
- There are 60 pink flowers.
- There are 105 roses.
- There are 20 flowers that are both pink and roses.
Step-by-step, we calculate the following:
1. Probability of a flower being pink:
[tex]\[ P(\text{Pink}) = \frac{\text{Total Pink Flowers}}{\text{Total Flowers}} = \frac{60}{315} \approx 0.1905 \][/tex]
2. Probability of a flower being a rose:
[tex]\[ P(\text{Rose}) = \frac{\text{Total Roses}}{\text{Total Flowers}} = \frac{105}{315} \approx 0.3333 \][/tex]
3. Probability of a flower being both pink and a rose:
[tex]\[ P(\text{Pink and Rose}) = \frac{\text{Pink Roses}}{\text{Total Flowers}} = \frac{20}{315} \approx 0.0635 \][/tex]
4. Check for independence between being pink and being a rose:
To check if the events are independent, we need to test if:
[tex]\[ P(\text{Pink and Rose}) = P(\text{Pink}) \times P(\text{Rose}) \][/tex]
We already know:
[tex]\[ P(\text{Pink}) \times P(\text{Rose}) = 0.1905 \times 0.3333 \approx 0.0635 \][/tex]
And we found:
[tex]\[ P(\text{Pink and Rose}) \approx 0.0635 \][/tex]
Since:
[tex]\[ P(\text{Pink and Rose}) = P(\text{Pink}) \times P(\text{Rose}) \][/tex]
It means that the events 'being pink' and 'being a rose' are independent.
Therefore, the correct statement is:
A. A flower being pink and a flower being a rose are independent.
