To address the question, we need to determine the relationship between the events "a flower is pink" and "a flower is a rose". We will check the independence of these two events using probabilities derived from the provided table.
Step-by-step solution:
1. Determine the total number of flowers in the table:
Total flowers = 315
2. Determine the number of pink flowers (regardless of type):
Total pink flowers = 60
3. Determine the number of roses (regardless of color):
Total roses = 105
4. Determine the number of flowers that are both pink and roses:
Pink roses = 20
5. Calculate the probability of a flower being pink (P(Pink)):
[tex]\( P(\text{Pink}) = \frac{\text{Total pink flowers}}{\text{Total flowers}} = \frac{60}{315} = 0.19047619047619047 \)[/tex]
6. Calculate the probability of a flower being a rose (P(Rose)):
[tex]\( P(\text{Rose}) = \frac{\text{Total roses}}{\text{Total flowers}} = \frac{105}{315} = 0.3333333333333333 \)[/tex]
7. Calculate the probability of a flower being both pink and a rose (P(Pink and Rose)):
[tex]\( P(\text{Pink and Rose}) = \frac{\text{Pink roses}}{\text{Total flowers}} = \frac{20}{315} = 0.06349206349206349 \)[/tex]
8. Calculate the product of the individual probabilities P(Pink) and P(Rose):
[tex]\( P(\text{Pink}) \times P(\text{Rose}) = 0.19047619047619047 \times 0.3333333333333333 = 0.06349206349206349 \)[/tex]
9. Compare P(Pink and Rose) with P(Pink) * P(Rose):
Since [tex]\( P(\text{Pink and Rose}) = P(\text{Pink}) \times P(\text{Rose}) = 0.06349206349206349 \)[/tex], this indicates that the events are independent.
10. Conclusion:
Since the calculated probabilities show that [tex]\( P(\text{Pink and Rose}) \)[/tex] is equal to [tex]\( P(\text{Pink}) \times P(\text{Rose}) \)[/tex], the events "a flower is pink" and "a flower is a rose" are independent events.
Therefore, the correct answer is:
A. A flower being pink and a flower being a rose are independent of each other.
