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?\no the events are independent because $p(red)cdot p(green)=p(red and green)$.\no the events are independent because $p(red)+p(green)=p(red and green)$.\no the events are dependent because $p(red)cdot p(green)\neq p(red and green)$.\no 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?\no the events are independent because $p(red)cdot p(green)=p(red and green)$.\no the events are independent because $p(red)+p(green)=p(red and green)$.\no the events are dependent because $p(red)cdot p(green)\neq p(red and green)$.\no the events are dependent because $p(red)+p(green)\neq p(red and green)$.

Answer

Answer:

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

Explanation:

Step1: Calculate $P(\text{red})\cdot P(\text{green})$

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

Step2: Compare with $P(\text{red and green})$

We know $P(\text{red and green})=\frac{1}{8}$. Since $P(\text{red})\cdot P(\text{green})=\frac{1}{8}$ and $P(\text{red and green})=\frac{1}{8}$, they are equal. For two events $A$ and $B$, if $P(A)\cdot P(B)=P(A\cap B)$, the events are independent.