Question

if events x and y are independent, what must be true? check all that apply.

\\(p(y | x) = 0\\)
\\(p(x | y) = 0\\)
\\(p(y | x) = p(y)\\)
\\(p(y | x) = p(x)\\)
\\(p(x | y) = p(y)\\)
\\(p(x | y) = p(x)\\)

Explanation:

Define independent events

Using the Independent Events knowledge point

By definition, two events \(X\) and \(Y\) are independent if the occurrence of one event does not affect the probability of the occurrence of the other. Mathematically, this is expressed as:

$$ P(X \cap Y) = P(X) \cdot P(Y) $$

Apply conditional probability formula

Using the Conditional Probability knowledge point

The conditional probability of event \(A\) given event \(B\) is defined as:

$$ P(A | B) = \frac{P(A \cap B)}{P(B)} \quad (\text{for } P(B) > 0) $$

Evaluate the given options

Using the Independent Events and Conditional Probability knowledge points

Substituting the independence condition \(P(X \cap Y) = P(X) \cdot P(Y)\) into the conditional probability formulas yields:

$$ P(Y | X) = \frac{P(X \cap Y)}{P(X)} = \frac{P(X) \cdot P(Y)}{P(X)} = P(Y) $$

$$ P(X | Y) = \frac{P(X \cap Y)}{P(Y)} = \frac{P(X) \cdot P(Y)}{P(Y)} = P(X) $$

Thus, the statements that must be true are \(P(Y | X) = P(Y)\) and \(P(X | Y) = P(X)\).

Answer:

*

$$P(Y | X) = 0$$

*

$$P(X | Y) = 0$$

• **

$$P(Y | X) = P(Y)$$

(Correct answer)**
*

$$P(Y | X) = P(X)$$

*

$$P(X | Y) = P(Y)$$

• **

$$P(X | Y) = P(X)$$

(Correct answer)**