Answer :
Let's analyze each of the provided statements using probabilities derived from the given table:
### Table Overview
The given table shows the distribution of flowers by color and type:
[tex]\[ \begin{array}{|c|c|c|c|c|} \hline \text{Type of Flower/Color} & \text{Red} & \text{Pink} & \text{Yellow} & \text{Total} \\ \hline \text{Rose} & 40 & 20 & 45 & 105 \\ \hline \text{Hibiscus} & 80 & 40 & 90 & 210 \\ \hline \text{Total} & 120 & 60 & 135 & 315 \\ \hline \end{array} \][/tex]
### Probabilities
- Total number of flowers: 315
- Total number of yellow flowers: 135
- Total number of roses: 105
- Total number of hibiscus: 210
- Total number of red flowers: 120
- Total number of pink flowers: 60
#### Calculation of Probabilities:
1. [tex]\( P(\text{flower is yellow}) \)[/tex]
[tex]\[ P(\text{flower is yellow}) = \frac{135}{315} \approx 0.42857142857142855 \][/tex]
2. [tex]\( P(\text{flower is a rose}) \)[/tex]
[tex]\[ P(\text{flower is a rose}) = \frac{105}{315} \approx 0.3333333333333333 \][/tex]
3. [tex]\( P(\text{flower is a hibiscus}) \)[/tex]
[tex]\[ P(\text{flower is a hibiscus}) = \frac{210}{315} \approx 0.6666666666666666 \][/tex]
4. [tex]\( P(\text{flower is red}) \)[/tex]
[tex]\[ P(\text{flower is red}) = \frac{120}{315} \approx 0.38095238095238093 \][/tex]
### Statements Assessment:
A. [tex]\( P(\text{flower is yellow | flower is a rose}) \neq P(\text{flower is yellow}) \)[/tex]
[tex]\[ P(\text{flower is yellow | flower is a rose}) = \frac{45}{105} \approx 0.42857142857142855 \][/tex]
Compare:
[tex]\[ P(\text{flower is yellow}) \approx 0.42857142857142855 \][/tex]
Both probabilities are equal, so this statement is false.
B. [tex]\( P(\text{flower is a hibiscus | color is red}) = P(\text{flower is a hibiscus}) \)[/tex]
[tex]\[ P(\text{flower is a hibiscus | color is red}) = \frac{80}{120} \approx 0.6666666666666666 \][/tex]
Compare:
[tex]\[ P(\text{flower is a hibiscus}) \approx 0.6666666666666666 \][/tex]
Both probabilities are equal, so this statement is true.
C. [tex]\( P(\text{flower is a rose | color is red}) = P(\text{flower is red}) \)[/tex]
[tex]\[ P(\text{flower is a rose | color is red}) = \frac{40}{120} \approx 0.3333333333333333 \][/tex]
Compare:
[tex]\[ P(\text{flower is red}) \approx 0.38095238095238093 \][/tex]
Probabilities are not equal, so this statement is false.
D. [tex]\( P(\text{flower is a hibiscus | color is pink}) \neq P(\text{flower is a hibiscus}) \)[/tex]
[tex]\[ P(\text{flower is a hibiscus | color is pink}) = \frac{40}{60} \approx 0.6666666666666666 \][/tex]
Compare:
[tex]\[ P(\text{flower is a hibiscus}) \approx 0.6666666666666666 \][/tex]
Both probabilities are equal, so this statement is false.
### Conclusion
The correct statement is B. [tex]\( P(\text{flower is a hibiscus | color is red}) = P(\text{flower is a hibiscus}) \)[/tex].
### Table Overview
The given table shows the distribution of flowers by color and type:
[tex]\[ \begin{array}{|c|c|c|c|c|} \hline \text{Type of Flower/Color} & \text{Red} & \text{Pink} & \text{Yellow} & \text{Total} \\ \hline \text{Rose} & 40 & 20 & 45 & 105 \\ \hline \text{Hibiscus} & 80 & 40 & 90 & 210 \\ \hline \text{Total} & 120 & 60 & 135 & 315 \\ \hline \end{array} \][/tex]
### Probabilities
- Total number of flowers: 315
- Total number of yellow flowers: 135
- Total number of roses: 105
- Total number of hibiscus: 210
- Total number of red flowers: 120
- Total number of pink flowers: 60
#### Calculation of Probabilities:
1. [tex]\( P(\text{flower is yellow}) \)[/tex]
[tex]\[ P(\text{flower is yellow}) = \frac{135}{315} \approx 0.42857142857142855 \][/tex]
2. [tex]\( P(\text{flower is a rose}) \)[/tex]
[tex]\[ P(\text{flower is a rose}) = \frac{105}{315} \approx 0.3333333333333333 \][/tex]
3. [tex]\( P(\text{flower is a hibiscus}) \)[/tex]
[tex]\[ P(\text{flower is a hibiscus}) = \frac{210}{315} \approx 0.6666666666666666 \][/tex]
4. [tex]\( P(\text{flower is red}) \)[/tex]
[tex]\[ P(\text{flower is red}) = \frac{120}{315} \approx 0.38095238095238093 \][/tex]
### Statements Assessment:
A. [tex]\( P(\text{flower is yellow | flower is a rose}) \neq P(\text{flower is yellow}) \)[/tex]
[tex]\[ P(\text{flower is yellow | flower is a rose}) = \frac{45}{105} \approx 0.42857142857142855 \][/tex]
Compare:
[tex]\[ P(\text{flower is yellow}) \approx 0.42857142857142855 \][/tex]
Both probabilities are equal, so this statement is false.
B. [tex]\( P(\text{flower is a hibiscus | color is red}) = P(\text{flower is a hibiscus}) \)[/tex]
[tex]\[ P(\text{flower is a hibiscus | color is red}) = \frac{80}{120} \approx 0.6666666666666666 \][/tex]
Compare:
[tex]\[ P(\text{flower is a hibiscus}) \approx 0.6666666666666666 \][/tex]
Both probabilities are equal, so this statement is true.
C. [tex]\( P(\text{flower is a rose | color is red}) = P(\text{flower is red}) \)[/tex]
[tex]\[ P(\text{flower is a rose | color is red}) = \frac{40}{120} \approx 0.3333333333333333 \][/tex]
Compare:
[tex]\[ P(\text{flower is red}) \approx 0.38095238095238093 \][/tex]
Probabilities are not equal, so this statement is false.
D. [tex]\( P(\text{flower is a hibiscus | color is pink}) \neq P(\text{flower is a hibiscus}) \)[/tex]
[tex]\[ P(\text{flower is a hibiscus | color is pink}) = \frac{40}{60} \approx 0.6666666666666666 \][/tex]
Compare:
[tex]\[ P(\text{flower is a hibiscus}) \approx 0.6666666666666666 \][/tex]
Both probabilities are equal, so this statement is false.
### Conclusion
The correct statement is B. [tex]\( P(\text{flower is a hibiscus | color is red}) = P(\text{flower is a hibiscus}) \)[/tex].
