Question

consider a situation in which ( p(x)=\frac{4}{5} ) and ( p(y)=\frac{1}{4} ). if ( p(x \text { and } y) ) is ( =\frac{1}{5} ), which best describes the events? they are independent because ( p(x) cdot p(y)=p(x \text { and } y) ). they are independent because ( p(x)+p(y)=p(x \text { and } y) ). they are dependent because ( p(x) cdot p(y)=p(x \text { and } y) ). they are dependent because ( p(x)+p(y)=p(x \text { and } y) ).

Explanation:

Step1: Calculate \(P(X)\cdot P(Y)\)

Given \(P(X)=\frac{4}{5}\) and \(P(Y)=\frac{1}{4}\), then \(P(X)\cdot P(Y)=\frac{4}{5}\times\frac{1}{4}=\frac{4\times1}{5\times4}=\frac{1}{5}\)

Step2: Compare with \(P(X\ and\ Y)\)

We know \(P(X\ and\ Y)=\frac{1}{5}\). Since \(P(X)\cdot P(Y) = P(X\ and\ Y)\), by the definition of independent events (if \(A\) and \(B\) are independent events, then \(P(A\cap B)=P(A)\cdot P(B)\)), we can check the relationship.
Also, \(P(X)+P(Y)=\frac{4}{5}+\frac{1}{4}=\frac{16 + 5}{20}=\frac{21}{20}
eq\frac{1}{5}\)

Answer:

They are independent because \(P(X)\cdot P(Y)=P(X\ and\ Y)\)