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: Recall Independence Formula

For two events \( X \) and \( Y \), they are independent if \( P(X \cap Y)=P(X)\cdot P(Y) \).

Step2: Calculate \( P(X)\cdot P(Y) \)

Given \( P(X)=\frac{4}{5} \) and \( P(Y)=\frac{1}{4} \), then \( P(X)\cdot P(Y)=\frac{4}{5}\times\frac{1}{4}=\frac{4}{20}=\frac{1}{5} \).

Step3: Compare with \( P(X \cap Y) \)

We know \( P(X \cap Y)=\frac{1}{5} \), so \( P(X)\cdot P(Y) = P(X \cap Y) \), which means the events are independent. Also, \( P(X)+P(Y)=\frac{4}{5}+\frac{1}{4}=\frac{16 + 5}{20}=\frac{21}{20}
eq\frac{1}{5} \), so the addition rule is not relevant here.

Answer:

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