To determine which statement is true, we need to analyze the independence of two events: a flower being pink and a flower being a rose. Let's go through the step-by-step solution to see which statement is accurate.
### Step-by-Step Analysis:
1. Total Number of Flowers:
The total number of flowers is 315.
2. Total Number of Roses:
The total number of roses is 105.
3. Total Number of Pink Flowers:
The total number of pink flowers is 60.
4. Number of Pink Roses:
There are 20 pink roses.
### Probability Calculations:
5. Probability a Flower is a Rose:
[tex]\[
P(\text{Rose}) = \frac{\text{Total Number of Roses}}{\text{Total Number of Flowers}} = \frac{105}{315} \approx 0.3333
\][/tex]
6. Probability a Flower is Pink:
[tex]\[
P(\text{Pink}) = \frac{\text{Total Number of Pink Flowers}}{\text{Total Number of Flowers}} = \frac{60}{315} \approx 0.1905
\][/tex]
7. Probability a Flower is both Pink and Rose:
[tex]\[
P(\text{Pink and Rose}) = \frac{\text{Number of Pink Roses}}{\text{Total Number of Flowers}} = \frac{20}{315} \approx 0.0635
\][/tex]
### Check for Independence:
To check if the two events (a flower being pink and a flower being a rose) are independent, we compare the probability of both events happening together with the product of the individual probabilities.
8. Calculate Product of Individual Probabilities:
[tex]\[
P(\text{Pink}) \times P(\text{Rose}) = 0.1905 \times 0.3333 \approx 0.0635
\][/tex]
9. Compare with Joint Probability:
[tex]\[
P(\text{Pink and Rose}) \approx 0.0635
\][/tex]
Both the joint probability and the product of individual probabilities are equal.
### Conclusion:
10. Since [tex]\( P(\text{Pink and Rose}) \approx P(\text{Pink}) \times P(\text{Rose}) \)[/tex], the events "being pink" and "being a rose" are independent of each other. Therefore, the correct statement is:
A. A flower being pink and a flower being a rose are independent of each other.
