Answer :
Let's analyze each statement one by one using the given data from the two-way table.
[tex]\[ \begin{tabular}{|c|c|c|c|c|} \hline Type of Flower/Color & Red & Pink & Yellow & Total \\ \hline Rose & 40 & 20 & 45 & 105 \\ \hline Hibiscus & 80 & 40 & 90 & 210 \\ \hline Total & 120 & 60 & 135 & 315 \\ \hline \end{tabular} \][/tex]
A. [tex]\(P(\text{flower is yellow} \mid \text{flower is rose}) \neq P(\text{flower is yellow})\)[/tex]
1. Probability that the flower is yellow given that it is a rose (conditional probability):
[tex]\[ P(\text{yellow} \mid \text{rose}) = \frac{\text{Number of yellow roses}}{\text{Total number of roses}} = \frac{45}{105} \approx 0.42857142857142855 \][/tex]
2. Probability that the flower is yellow (marginal probability):
[tex]\[ P(\text{yellow}) = \frac{\text{Total number of yellow flowers}}{\text{Total number of flowers}} = \frac{135}{315} \approx 0.42857142857142855 \][/tex]
Comparing these probabilities, we see that:
[tex]\[ P(\text{yellow} \mid \text{rose}) = P(\text{yellow}) \][/tex]
Therefore, the statement [tex]\(P(\text{flower is yellow} \mid \text{flower is rose}) \neq P(\text{flower is yellow})\)[/tex] is False.
B. [tex]\(P(\text{flower is hibiscus} \mid \text{color is red}) = P(\text{flower is hibiscus})\)[/tex]
1. Probability that the flower is hibiscus given that it is red (conditional probability):
[tex]\[ P(\text{hibiscus} \mid \text{red}) = \frac{\text{Number of red hibiscus flowers}}{\text{Total number of red flowers}} = \frac{80}{120} \approx 0.6666666666666666 \][/tex]
2. Probability that the flower is hibiscus (marginal probability):
[tex]\[ P(\text{hibiscus}) = \frac{\text{Total number of hibiscus flowers}}{\text{Total number of flowers}} = \frac{210}{315} \approx 0.6666666666666666 \][/tex]
Comparing these probabilities, we see that:
[tex]\[ P(\text{hibiscus} \mid \text{red}) = P(\text{hibiscus}) \][/tex]
Therefore, the statement [tex]\(P(\text{flower is hibiscus} \mid \text{color is red}) = P(\text{flower is hibiscus})\)[/tex] is True.
C. [tex]\(P(\text{flower is rose} \mid \text{color is red}) = P(\text{flower is red})\)[/tex]
1. Probability that the flower is rose given that it is red (conditional probability):
[tex]\[ P(\text{rose} \mid \text{red}) = \frac{\text{Number of red roses}}{\text{Total number of red flowers}} = \frac{40}{120} \approx 0.3333333333333333 \][/tex]
2. Probability that the flower is red (marginal probability):
[tex]\[ P(\text{red}) = \frac{\text{Total number of red flowers}}{\text{Total number of flowers}} = \frac{120}{315} \approx 0.38095238095238093 \][/tex]
Comparing these probabilities, we see that:
[tex]\[ P(\text{rose} \mid \text{red}) \neq P(\text{red}) \][/tex]
Therefore, the statement [tex]\(P(\text{flower is rose} \mid \text{color is red}) = P(\text{flower is red})\)[/tex] is False.
D. [tex]\(P(\text{flower is hibiscus} \mid \text{color is pink}) \neq P(\text{flower is hibiscus})\)[/tex]
1. Probability that the flower is hibiscus given that it is pink (conditional probability):
[tex]\[ P(\text{hibiscus} \mid \text{pink}) = \frac{\text{Number of pink hibiscus flowers}}{(\text{Number of pink hibiscus flowers} + \text{Number of pink roses})} = \frac{40}{(40 + 20)} = \frac{40}{60} \approx 0.6666666666666666 \][/tex]
2. Probability that the flower is hibiscus (marginal probability):
[tex]\[ P(\text{hibiscus}) = \frac{\text{Total number of hibiscus flowers}}{\text{Total number of flowers}} = \frac{210}{315} \approx 0.6666666666666666 \][/tex]
Comparing these probabilities, we see that:
[tex]\[ P(\text{hibiscus} \mid \text{pink}) = P(\text{hibiscus}) \][/tex]
Therefore, the statement [tex]\(P(\text{flower is hibiscus} \mid \text{color is pink}) \neq P(\text{flower is hibiscus})\)[/tex] is False.
In conclusion, the only true statement among the given options is:
[tex]\[ \text{B. } P(\text{flower is hibiscus} \mid \text{color is red}) = P(\text{flower is hibiscus}) \][/tex]
[tex]\[ \begin{tabular}{|c|c|c|c|c|} \hline Type of Flower/Color & Red & Pink & Yellow & Total \\ \hline Rose & 40 & 20 & 45 & 105 \\ \hline Hibiscus & 80 & 40 & 90 & 210 \\ \hline Total & 120 & 60 & 135 & 315 \\ \hline \end{tabular} \][/tex]
A. [tex]\(P(\text{flower is yellow} \mid \text{flower is rose}) \neq P(\text{flower is yellow})\)[/tex]
1. Probability that the flower is yellow given that it is a rose (conditional probability):
[tex]\[ P(\text{yellow} \mid \text{rose}) = \frac{\text{Number of yellow roses}}{\text{Total number of roses}} = \frac{45}{105} \approx 0.42857142857142855 \][/tex]
2. Probability that the flower is yellow (marginal probability):
[tex]\[ P(\text{yellow}) = \frac{\text{Total number of yellow flowers}}{\text{Total number of flowers}} = \frac{135}{315} \approx 0.42857142857142855 \][/tex]
Comparing these probabilities, we see that:
[tex]\[ P(\text{yellow} \mid \text{rose}) = P(\text{yellow}) \][/tex]
Therefore, the statement [tex]\(P(\text{flower is yellow} \mid \text{flower is rose}) \neq P(\text{flower is yellow})\)[/tex] is False.
B. [tex]\(P(\text{flower is hibiscus} \mid \text{color is red}) = P(\text{flower is hibiscus})\)[/tex]
1. Probability that the flower is hibiscus given that it is red (conditional probability):
[tex]\[ P(\text{hibiscus} \mid \text{red}) = \frac{\text{Number of red hibiscus flowers}}{\text{Total number of red flowers}} = \frac{80}{120} \approx 0.6666666666666666 \][/tex]
2. Probability that the flower is hibiscus (marginal probability):
[tex]\[ P(\text{hibiscus}) = \frac{\text{Total number of hibiscus flowers}}{\text{Total number of flowers}} = \frac{210}{315} \approx 0.6666666666666666 \][/tex]
Comparing these probabilities, we see that:
[tex]\[ P(\text{hibiscus} \mid \text{red}) = P(\text{hibiscus}) \][/tex]
Therefore, the statement [tex]\(P(\text{flower is hibiscus} \mid \text{color is red}) = P(\text{flower is hibiscus})\)[/tex] is True.
C. [tex]\(P(\text{flower is rose} \mid \text{color is red}) = P(\text{flower is red})\)[/tex]
1. Probability that the flower is rose given that it is red (conditional probability):
[tex]\[ P(\text{rose} \mid \text{red}) = \frac{\text{Number of red roses}}{\text{Total number of red flowers}} = \frac{40}{120} \approx 0.3333333333333333 \][/tex]
2. Probability that the flower is red (marginal probability):
[tex]\[ P(\text{red}) = \frac{\text{Total number of red flowers}}{\text{Total number of flowers}} = \frac{120}{315} \approx 0.38095238095238093 \][/tex]
Comparing these probabilities, we see that:
[tex]\[ P(\text{rose} \mid \text{red}) \neq P(\text{red}) \][/tex]
Therefore, the statement [tex]\(P(\text{flower is rose} \mid \text{color is red}) = P(\text{flower is red})\)[/tex] is False.
D. [tex]\(P(\text{flower is hibiscus} \mid \text{color is pink}) \neq P(\text{flower is hibiscus})\)[/tex]
1. Probability that the flower is hibiscus given that it is pink (conditional probability):
[tex]\[ P(\text{hibiscus} \mid \text{pink}) = \frac{\text{Number of pink hibiscus flowers}}{(\text{Number of pink hibiscus flowers} + \text{Number of pink roses})} = \frac{40}{(40 + 20)} = \frac{40}{60} \approx 0.6666666666666666 \][/tex]
2. Probability that the flower is hibiscus (marginal probability):
[tex]\[ P(\text{hibiscus}) = \frac{\text{Total number of hibiscus flowers}}{\text{Total number of flowers}} = \frac{210}{315} \approx 0.6666666666666666 \][/tex]
Comparing these probabilities, we see that:
[tex]\[ P(\text{hibiscus} \mid \text{pink}) = P(\text{hibiscus}) \][/tex]
Therefore, the statement [tex]\(P(\text{flower is hibiscus} \mid \text{color is pink}) \neq P(\text{flower is hibiscus})\)[/tex] is False.
In conclusion, the only true statement among the given options is:
[tex]\[ \text{B. } P(\text{flower is hibiscus} \mid \text{color is red}) = P(\text{flower is hibiscus}) \][/tex]
