Question

a bag contains a variety of different - colored marbles. if $p(\text{red}) = \frac{1}{2}$, $p(\text{green}) = \frac{1}{4}$, and $p(\text{red and green}) = \frac{1}{8}$, which statement is true?\
\
\bigcirc the events are independent because $p(\text{red})\cdot p(\text{green}) = p(\text{red and green})$.\
\bigcirc the events are independent because $p(\text{red}) + p(\text{green}) = p(\text{red and green})$.\
\bigcirc the events are dependent because $p(\text{red})\cdot p(\text{green}) \
eq p(\text{red and green})$.\
\bigcirc the events are dependent because $p(\text{red}) + p(\text{green}) \
eq p(\text{red and green})$.

Explanation:

Step1: Recall independence rule

For two events \( A \) and \( B \), they are independent if \( P(A \cap B)=P(A)\cdot P(B) \).

Step2: Calculate \( P(\text{red})\cdot P(\text{green}) \)

Given \( P(\text{red}) = \frac{1}{2} \), \( P(\text{green})=\frac{1}{4} \).
So \( P(\text{red})\cdot P(\text{green})=\frac{1}{2}\cdot\frac{1}{4}=\frac{1}{8} \).

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

Given \( P(\text{red and green})=\frac{1}{8} \).
Since \( P(\text{red})\cdot P(\text{green}) = P(\text{red and green}) \), the events are independent.

Answer:

The events are independent because \( P(\text{red}) \cdot P(\text{green}) = P(\text{red and green}) \). (The first option)