Let's analyze the statements given the data in the two-way table.
We have the following information:
- Number of red roses: 40
- Number of pink roses: 20
- Number of yellow roses: 45
- Number of red hibiscus: 80
- Number of pink hibiscus: 40
- Number of yellow hibiscus: 90
- Total red flowers: 120
- Total pink flowers: 60
- Total yellow flowers: 135
- Total roses: 105
- Total hibiscus: 210
- Total flowers: 315
We'll evaluate each statement step-by-step.
### Statement A: [tex]\( P(\text{flower is yellow} \mid \text{flower is rose}) \neq P(\text{flower is yellow}) \)[/tex]
1. Calculate [tex]\( P(\text{flower is yellow} \mid \text{flower is rose}) \)[/tex]:
[tex]\( P(\text{yellow} \mid \text{rose}) = \frac{\text{Number of yellow roses}}{\text{Total number of roses}} = \frac{45}{105} = \frac{3}{7} \)[/tex]
2. Calculate [tex]\( P(\text{flower is yellow}) \)[/tex]:
[tex]\( P(\text{yellow}) = \frac{\text{Total number of yellow flowers}}{\text{Total number of flowers}} = \frac{135}{315} = \frac{9}{21} = \frac{3}{7} \)[/tex]
We find that [tex]\( P(\text{yellow} \mid \text{rose}) = \frac{3}{7} \)[/tex] and [tex]\( P(\text{yellow}) = \frac{3}{7} \)[/tex]. Therefore, Statement A is false.
### Statement B: [tex]\( P(\text{flower is hibiscus} \mid \text{color is red}) = P(\text{flower is hibiscus}) \)[/tex]
1. Calculate [tex]\( P(\text{flower is hibiscus} \mid \text{color is red}) \)[/tex]:
[tex]\( P(\text{hibiscus} \mid \text{red}) = \frac{\text{Number of red hibiscus}}{\text{Total number of red flowers}} = \frac{80}{120} = \frac{8}{12} = \frac{2}{3} \)[/tex]
2. Calculate [tex]\( P(\text{flower is hibiscus}) \)[/tex]:
[tex]\( P(\text{hibiscus}) = \frac{\text{Total number of hibiscus}}{\text{Total number of flowers}} = \frac{210}{315} = \frac{2}{3} \)[/tex]
We find that [tex]\( P(\text{hibiscus} \mid \text{red}) = \frac{2}{3} \)[/tex] and [tex]\( P(\text{hibiscus}) = \frac{2}{3} \)[/tex]. Therefore, Statement B is true.
### Statement C: [tex]\( P(\text{flower is rose} \mid \text{color is red}) = P(\text{flower is red}) \)[/tex]
1. Calculate [tex]\( P(\text{flower is rose} \mid \text{color is red}) \)[/tex]:
[tex]\( P(\text{rose} \mid \text{red}) = \frac{\text{Number of red roses}}{\text{Total number of red flowers}} = \frac{40}{120} = \frac{1}{3} \)[/tex]
2. Calculate [tex]\( P(\text{flower is red}) \)[/tex]:
[tex]\( P(\text{red}) = \frac{\text{Total number of red flowers}}{\text{Total number of flowers}} = \frac{120}{315} = \frac{8}{21} \)[/tex]
We find that [tex]\( P(\text{rose} \mid \text{red}) = \frac{1}{3} \neq \frac{8}{21} \)[/tex]. Therefore, Statement C is false.
### Statement D: [tex]\( P(\text{flower is hibiscus} \mid \text{color is pink}) \neq P(\text{flower is hibiscus}) \)[/tex]
1. Calculate [tex]\( P(\text{flower is hibiscus} \mid \text{color is pink}) \)[/tex]:
[tex]\( P(\text{hibiscus} \mid \text{pink}) = \frac{\text{Number of pink hibiscus}}{\text{Total number of pink flowers}} = \frac{40}{60} = \frac{2}{3} \)[/tex]
2. Calculate [tex]\( P(\text{flower is hibiscus}) \)[/tex] (as done previously):
[tex]\( P(\text{hibiscus}) = \frac{210}{315} = \frac{2}{3} \)[/tex]
We find that [tex]\( P(\text{hibiscus} \mid \text{pink}) = \frac{2}{3} \)[/tex] and [tex]\( P(\text{hibiscus}) = \frac{2}{3} \)[/tex]. Therefore, Statement D is false.
Thus, the correct statement is:
B. [tex]\( P(\text{flower is hibiscus} \mid \text{color is red}) = P(\text{flower is hibiscus}) \)[/tex]
