Answer :
Let's solve the problem by determining if the events "being a pink flower" and "being a rose" are independent.
Step 1: Define the given data from the table.
- Number of pink roses = 20
- Total number of roses = 105
- Total number of pink flowers = 60
- Total number of flowers = 315
Step 2: Calculate the probability of each event.
- [tex]\( P(\text{pink}) \)[/tex] is the probability of choosing a pink flower:
[tex]\[ P(\text{pink}) = \frac{\text{total number of pink flowers}}{\text{total number of flowers}} = \frac{60}{315} \][/tex]
- [tex]\( P(\text{rose}) \)[/tex] is the probability of choosing a rose:
[tex]\[ P(\text{rose}) = \frac{\text{total number of roses}}{\text{total number of flowers}} = \frac{105}{315} \][/tex]
- [tex]\( P(\text{pink and rose}) \)[/tex] is the probability of choosing a flower that is both pink and a rose:
[tex]\[ P(\text{pink and rose}) = \frac{\text{number of pink roses}}{\text{total number of flowers}} = \frac{20}{315} \][/tex]
Step 3: Determine if the events are independent.
To check if the events are independent, we need to verify if:
[tex]\[ P(\text{pink and rose}) = P(\text{pink}) \times P(\text{rose}) \][/tex]
Calculate [tex]\( P(\text{pink}) \times P(\text{rose}) \)[/tex]:
[tex]\[ P(\text{pink}) \times P(\text{rose}) = \left(\frac{60}{315}\right) \times \left(\frac{105}{315}\right) \][/tex]
Step 4: Compare [tex]\( P(\text{pink and rose}) \)[/tex] and [tex]\( P(\text{pink}) \times P(\text{rose}) \)[/tex].
- [tex]\( P(\text{pink and rose}) = \frac{20}{315} \)[/tex]
- [tex]\( P(\text{pink}) \times P(\text{rose}) = \left(\frac{60}{315}\right) \times \left(\frac{105}{315}\right) = \frac{6300}{99225} = \frac{20}{315} \)[/tex]
Since [tex]\( P(\text{pink and rose}) \)[/tex] is indeed equal to [tex]\( P(\text{pink}) \times P(\text{rose}) \)[/tex], the events "being a pink flower" and "being a rose" are independent.
Conclusion: The correct answer is:
A. A flower being pink and a flower being a rose are independent of each other.
Step 1: Define the given data from the table.
- Number of pink roses = 20
- Total number of roses = 105
- Total number of pink flowers = 60
- Total number of flowers = 315
Step 2: Calculate the probability of each event.
- [tex]\( P(\text{pink}) \)[/tex] is the probability of choosing a pink flower:
[tex]\[ P(\text{pink}) = \frac{\text{total number of pink flowers}}{\text{total number of flowers}} = \frac{60}{315} \][/tex]
- [tex]\( P(\text{rose}) \)[/tex] is the probability of choosing a rose:
[tex]\[ P(\text{rose}) = \frac{\text{total number of roses}}{\text{total number of flowers}} = \frac{105}{315} \][/tex]
- [tex]\( P(\text{pink and rose}) \)[/tex] is the probability of choosing a flower that is both pink and a rose:
[tex]\[ P(\text{pink and rose}) = \frac{\text{number of pink roses}}{\text{total number of flowers}} = \frac{20}{315} \][/tex]
Step 3: Determine if the events are independent.
To check if the events are independent, we need to verify if:
[tex]\[ P(\text{pink and rose}) = P(\text{pink}) \times P(\text{rose}) \][/tex]
Calculate [tex]\( P(\text{pink}) \times P(\text{rose}) \)[/tex]:
[tex]\[ P(\text{pink}) \times P(\text{rose}) = \left(\frac{60}{315}\right) \times \left(\frac{105}{315}\right) \][/tex]
Step 4: Compare [tex]\( P(\text{pink and rose}) \)[/tex] and [tex]\( P(\text{pink}) \times P(\text{rose}) \)[/tex].
- [tex]\( P(\text{pink and rose}) = \frac{20}{315} \)[/tex]
- [tex]\( P(\text{pink}) \times P(\text{rose}) = \left(\frac{60}{315}\right) \times \left(\frac{105}{315}\right) = \frac{6300}{99225} = \frac{20}{315} \)[/tex]
Since [tex]\( P(\text{pink and rose}) \)[/tex] is indeed equal to [tex]\( P(\text{pink}) \times P(\text{rose}) \)[/tex], the events "being a pink flower" and "being a rose" are independent.
Conclusion: The correct answer is:
A. A flower being pink and a flower being a rose are independent of each other.
