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:

When two events \(X\) and \(Y\) are independent, the occurrence of one event does not affect the probability of the occurrence of the other event. By the definition of conditional probability \(P(A|B)=\frac{P(A\cap B)}{P(B)}\), and for independent events \(P(A\cap B) = P(A)\times P(B)\).

For \(P(Y|X)\), we have \(P(Y|X)=\frac{P(X\cap Y)}{P(X)}\). Since \(P(X\cap Y)=P(X)\times P(Y)\) (because \(X\) and \(Y\) are independent), then \(P(Y|X)=\frac{P(X)\times P(Y)}{P(X)} = P(Y)\).

Similarly, for \(P(X|Y)\), \(P(X|Y)=\frac{P(X\cap Y)}{P(Y)}\). Substituting \(P(X\cap Y)=P(X)\times P(Y)\) (due to independence), we get \(P(X|Y)=\frac{P(X)\times P(Y)}{P(Y)}=P(X)\)

Answer:

\(P(Y | X)=P(Y)\), \(P(X | Y)=P(X)\)