Question

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

Explanation:

Step1: Recall the independence - formula

For two events \(A\) and \(B\), if they are independent, \(P(A)\times P(B)=P(A\cap B)\). Here \(A\) is the event of getting a red marble (\(P(\text{red})=\frac{1}{2}\)) and \(B\) is the event of getting a green marble (\(P(\text{green})=\frac{1}{4}\)), and \(P(\text{red and green}) = P(\text{red}\cap\text{green})=\frac{1}{8}\).

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

\[P(\text{red})\times P(\text{green})=\frac{1}{2}\times\frac{1}{4}=\frac{1}{8}\]

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

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

Answer:

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