A standard six-sided die is rolled. Which of the following calculations is used to determine if the events "roll a number greater than 4" and "roll an even number" are independent?

A. [tex]\frac{\frac{2}{6}}{\frac{3}{6}} \neq \frac{3}{6}[/tex]

B. [tex]\frac{\frac{3}{6}}{\frac{1}{6}} \neq \frac{2}{6}[/tex]

C. [tex]\frac{\frac{3}{6}}{\frac{2}{6}} \neq \frac{2}{6}[/tex]

D. [tex]\frac{\frac{1}{6}}{\frac{2}{6}} \neq \frac{3}{6}[/tex]



Answer :

To determine if two events [tex]\( A \)[/tex] (rolling a number greater than 4) and [tex]\( B \)[/tex] (rolling an even number) are independent, we need to check if the probability of their intersection [tex]\( P(A \cap B) \)[/tex] equals the product of their individual probabilities [tex]\( P(A) \times P(B) \)[/tex].

First, let’s define the probabilities:

1. [tex]\( P(A) \)[/tex] is the probability of rolling a number greater than 4. On a six-sided die, the possible outcomes greater than 4 are 5 and 6. Therefore:
[tex]\[ P(A) = \frac{2}{6} \][/tex]

2. [tex]\( P(B) \)[/tex] is the probability of rolling an even number. The even numbers on a six-sided die are 2, 4, and 6. Therefore:
[tex]\[ P(B) = \frac{3}{6} \][/tex]

3. [tex]\( P(A \cap B) \)[/tex] is the probability of both events happening (rolling a number that is both greater than 4 and even). The only number that fits this criterion is 6. Therefore:
[tex]\[ P(A \cap B) = \frac{1}{6} \][/tex]

For events [tex]\( A \)[/tex] and [tex]\( B \)[/tex] to be independent, the following condition must hold:
[tex]\[ P(A \cap B) = P(A) \times P(B) \][/tex]

First, let’s compute [tex]\( P(A) \times P(B) \)[/tex]:
[tex]\[ P(A) \times P(B) = \frac{2}{6} \times \frac{3}{6} = \frac{6}{36} = \frac{1}{6} \][/tex]

Therefore, [tex]\( P(A \cap B) = P(A) \times P(B) \)[/tex] holds true.

Now, let’s double-check with the given question by examining each option for the necessary calculations:

1. [tex]\(\frac{\frac{2}{6}}{\frac{3}{6}} \neq \frac{3}{6}\)[/tex]
[tex]\[ \frac{\frac{2}{6}}{\frac{3}{6}} = \frac{2}{3} \quad \text{and} \quad \frac{3}{6} = 0.5 \quad \Rightarrow \quad \frac{2}{3} \neq 0.5 \][/tex]

2. [tex]\(\frac{\frac{3}{6}}{\frac{1}{6}} \neq \frac{2}{6}\)[/tex]
[tex]\[ \frac{\frac{3}{6}}{\frac{1}{6}} = 3 \quad \text{and} \quad \frac{2}{6} = 0.333... \quad \Rightarrow \quad 3 \neq 0.333... \][/tex]

3. [tex]\(\frac{\frac{3}{6}}{\frac{2}{6}} \neq \frac{2}{6}\)[/tex]
[tex]\[ \frac{\frac{3}{6}}{\frac{2}{6}} = \frac{3}{2} = 1.5 \quad \text{and} \quad \frac{2}{6} = 0.333... \quad \Rightarrow \quad 1.5 \neq 0.333... \][/tex]

4. [tex]\(\frac{\frac{1}{6}}{\frac{2}{6}} \neq \frac{3}{6}\)[/tex]
[tex]\[ \frac{\frac{1}{6}}{\frac{2}{6}} = \frac{1}{2} = 0.5 \quad \text{and} \quad\frac{3}{6} = 0.5 \quad \Rightarrow \quad 0.5 = 0.5 \][/tex]

Based on the comparisons, option [tex]\( \frac{\frac{1}{6}}{\frac{2}{6}} \neq \frac{3}{6} \)[/tex] correctly represents the calculation used to determine the independence condition that fails in this given context.

Therefore, the correct answer is:
[tex]\[ \frac{\frac{1}{6}}{\frac{2}{6}} \neq \frac{3}{6} \][/tex]