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 the independence rule for events

For two events \( A \) and \( B \), they are independent if \( P(A \cap B)=P(A)\times P(B) \). Here, event \( A \) is "red" and event \( B \) is "green", so we check if \( P(\text{red and green}) = P(\text{red})\times P(\text{green}) \).

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

Given \( P(\text{red})=\frac{1}{2} \) and \( P(\text{green})=\frac{1}{4} \), then \( P(\text{red})\times P(\text{green})=\frac{1}{2}\times\frac{1}{4}=\frac{1}{8} \).

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

We know \( P(\text{red and green})=\frac{1}{8} \), so \( P(\text{red})\times P(\text{green}) = P(\text{red and green}) \). By the definition of independent events, this means 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)