a bag contains a variety of different - colored marbles. if $p(red)=\frac{1}{2}$, $p(green)=\frac{1}{4}$…

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 $p(red)cdot p(green)\neq p(red and green)$. the events are dependent because $p(red)cdot p(green)\neq p(red and green)$. the events are dependent because $p(red)+p(green)\neq p(red and green)$.

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 $p(red)cdot p(green)\neq p(red and green)$. the events are dependent because $p(red)cdot p(green)\neq p(red and green)$. the events are dependent because $p(red)+p(green)\neq p(red and green)$.

Answer

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})).