Question

a box contains different - colored marbles. if $p(\text{blue}) = \frac{1}{4}$, $p(\text{green}) = \frac{1}{4}$, and $p(\text{blue and green}) = \frac{1}{12}$, which statement is true?
the events are independent because $p(\text{blue}) \cdot p(\text{green}) = p(\text{blue and green})$.
the events are independent because $p(\text{blue}) \cdot p(\text{green}) \
eq p(\text{blue and green})$.
the events are dependent because $p(\text{blue}) \cdot p(\text{green}) = p(\text{blue and green})$.
the events are dependent because $p(\text{blue}) \cdot p(\text{green}) \
eq p(\text{blue and green})$.

Explanation:

Step1: Recall the independence formula

For two events \( A \) and \( B \), they are independent if \( P(A \cap B)=P(A)\times P(B) \). Here, \( A \) is blue, \( B \) is green.

Step2: Calculate \( P(\text{blue})\times P(\text{green}) \)

Given \( P(\text{blue})=\frac{1}{4} \), \( P(\text{green})=\frac{1}{4} \), so \( P(\text{blue})\times P(\text{green})=\frac{1}{4}\times\frac{1}{4}=\frac{1}{16} \).

Step3: Compare with \( P(\text{blue and green}) \)

Given \( P(\text{blue and green})=\frac{1}{12} \). Since \( \frac{1}{16}
eq\frac{1}{12} \), i.e., \( P(\text{blue})\times P(\text{green})
eq P(\text{blue and green}) \), the events are dependent.

Answer:

The events are dependent because \( P(\text{blue}) \cdot P(\text{green})
eq P(\text{blue and green}) \). (Assuming the last option has a typo and should be "The events are dependent because \( P(\text{blue}) \cdot P(\text{green})
eq P(\text{blue and green}) \)")