Question

a box contains different - colored marbles. if ( p(\text{blue})=\frac{1}{4},p(\text{green})=\frac{1}{4} ), and ( p(\text{blue and green})=\frac{1}{12} ), which statement is true?
the events are independent because ( p(\text{blue})cdot p(\text{green}) = p(\text{blue and green}) ).
the events are independent because ( p(\text{blue})cdot p(\text{green})
eq p(\text{blue and green}) ).
the events are dependent because ( p(\text{blue})cdot p(\text{green}) = p(\text{blue and green}) ).
the events are dependent because ( p(\text{blue})cdot p(\text{green})
eq p(\text{blue and green}) ).

Explanation:

Step1: Calculate \(P(blue)\cdot P(green)\)

Given \(P(blue)=\frac{1}{4}\) and \(P(green)=\frac{1}{4}\), then \(P(blue)\cdot P(green)=\frac{1}{4}\times\frac{1}{4}=\frac{1}{16}\)

Step2: Compare \(P(blue)\cdot P(green)\) with \(P(blue\ and\ green)\)

We know \(P(blue\ and\ green)=\frac{1}{12}\). Since \(\frac{1}{16}
eq\frac{1}{12}\) (because \(16
eq12\) when denominators are compared for the fractions \(\frac{1}{16}\) and \(\frac{1}{12}\)), and for two events \(A\) and \(B\), if \(P(A)\cdot P(B)
eq P(A\cap B)\), the events are dependent.

Answer:

The events are dependent because \(P(blue)\cdot P(green)
eq P(blue\ and\ green)\).