Select the correct answer.

Based on the data in this two-way table, which statement is true?

\begin{tabular}{|c|c|c|c|c|}
\hline Type of Flower & 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}

A. A flower being pink and a flower being a rose are independent of each other.
B. A flower being pink is dependent on a flower being a rose.
C. A flower being a rose is dependent on a flower being pink.
D. A flower being pink and a flower being a rose are the same.



Answer :

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.