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)

Answer 1

Correct choices:

- P(Y | X) = P(Y)
- P(X | Y) = P(X)

Reason: Independence means P(X ∩ Y) = P(X)P(Y). If P(X) > 0 then P(Y | X) = P(X ∩ Y)/P(X) = P(Y). If P(Y) > 0 then P(X | Y) = P(X ∩ Y)/P(Y) = P(X). The other equalities need not hold (they can be true in special cases, e.g. if one probability is 0 or P(X)=P(Y), but are not generally implied by independence).