To determine which statement is true based on the data in the two-way table, we need to assess whether the events "flower being pink" and "flower being a rose" are independent.
1. Calculate the probabilities:
- The total number of flowers is 315.
- The total number of pink flowers is 60.
- The total number of roses is 105.
- The number of flowers that are both pink and roses is 20.
Let's denote:
- [tex]\( P(A) \)[/tex] as the probability that a flower is pink.
- [tex]\( P(B) \)[/tex] as the probability that a flower is a rose.
- [tex]\( P(A \cap B) \)[/tex] as the probability that a flower is both pink and a rose.
2. Determine individual probabilities:
- [tex]\( P(A) = \frac{\text{Number of pink flowers}}{\text{Total number of flowers}} = \frac{60}{315} \)[/tex]
- [tex]\( P(B) = \frac{\text{Number of roses}}{\text{Total number of flowers}} = \frac{105}{315} \)[/tex]
- [tex]\( P(A \cap B) = \frac{\text{Number of pink roses}}{\text{Total number of flowers}} = \frac{20}{315} \)[/tex]
3. Check for independence:
Two events [tex]\( A \)[/tex] and [tex]\( B \)[/tex] are independent if and only if [tex]\( P(A \cap B) = P(A) \cdot P(B) \)[/tex].
- Calculate [tex]\( P(A) \cdot P(B) \)[/tex]:
[tex]\[ P(A) \cdot P(B) = \left( \frac{60}{315} \right) \cdot \left( \frac{105}{315} \right) \][/tex]
Simplify the fractions:
[tex]\[ P(A) \cdot P(B) = \left( \frac{4}{21} \right) \cdot \left( \frac{1}{3} \right) \][/tex]
[tex]\[ P(A) \cdot P(B) = \frac{4}{63} \][/tex]
- Compare [tex]\( P(A \cap B) \)[/tex] with [tex]\( P(A) \cdot P(B) \)[/tex]:
[tex]\[ P(A \cap B) = \frac{20}{315} = \frac{4}{63} \][/tex]
4. Conclusion:
Since the computed [tex]\( P(A \cap B) \)[/tex] (i.e., [tex]\(\frac{4}{63}\)[/tex]) is equal to [tex]\( P(A) \cdot P(B) \)[/tex] (i.e., [tex]\(\frac{4}{63}\)[/tex]), the events "flower being pink" and "flower being a rose" are independent.
Given this analysis, the correct statement is:
A. A flower being pink and a flower being a rose are independent of each other.
