Question

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

Explanation:

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

$P(X)\cdot P(Y)=\frac{4}{5}\times\frac{1}{4}=\frac{4\times1}{5\times4}=\frac{1}{5}$

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

We know \(P(X\ and\ Y)=\frac{1}{5}\), and since \(P(X)\cdot P(Y) = P(X\ and\ Y)\), the events are independent. The condition for two - events \(X\) and \(Y\) to be independent is \(P(X)\cdot P(Y)=P(X\ and\ Y)\).

Answer:

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