Question

select the correct answer. based on the data in this two - way table, which statement is true? 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.
type of flower\color\trose\thibiscus\ttotal
red\t40\t80\t120
pink\t20\t40\t60
yellow\t45\t90\t135
total\t105\t210\t315

Explanation:

Step1: Recall the concept of independence

Two events \(A\) and \(B\) are independent if \(P(A\cap B)=P(A)\times P(B)\). Let event \(A\) be a flower being pink and event \(B\) be a flower being a rose.

Step2: Calculate probabilities

The probability of a flower being pink \(P(\text{pink})=\frac{60 + 20}{315}=\frac{80}{315}=\frac{16}{63}\). The probability of a flower being a rose \(P(\text{rose})=\frac{40+20 + 45+10}{315}=\frac{115}{315}=\frac{23}{63}\). The probability of a flower being pink and a rose \(P(\text{pink}\cap\text{rose})=\frac{20}{315}=\frac{4}{63}\). And \(P(\text{pink})\times P(\text{rose})=\frac{16}{63}\times\frac{23}{63}=\frac{368}{3969}
eq\frac{4}{63}\), so they are dependent.

Step3: Analyze each option

• Option A: Since \(P(\text{pink}\cap\text{rose})

eq P(\text{pink})\times P(\text{rose})\), a flower being pink and a flower being a rose are not independent, so A is false.

• Option B: As calculated above, a flower being pink is dependent on a flower being a rose, so B is true.
• Option C: The correct statement is that a flower being a rose is dependent on a flower being pink, not the other - way - around as stated in a wrong structure in C, so C is false.
• Option D: They are not the same, so D is false.

Answer:

B. A flower being pink is dependent on a flower being a rose.