Question

determining if events are independent or dependent
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?
the events are independent because ( p(\text{red})cdot p(\text{green}) = p(\text{red and green}) ).
the events are independent because ( p(\text{red})+p(\text{green}) = p(\text{red and green}) ).
the events are dependent because ( p(\text{red})cdot p(\text{green})
eq p(\text{red and green}) ).
the events are dependent because ( p(\text{red})+p(\text{green})
eq p(\text{red and green}) ).

Explanation:

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

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

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

We know \(P(\text{red and green})=\frac{1}{8}\). Since \(P(\text{red})\cdot P(\text{green}) = P(\text{red and green})\), by the definition of independent events (if \(A\) and \(B\) are events, \(A\) and \(B\) are independent if \(P(A\cap B)=P(A)\cdot P(B)\))

Answer:

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