zero correlation does not imply independence
This is one of the most important and most often-violated facts in all of probability. It is tempting to read 'correlation zero' as 'no relationship at all', and then to treat the two variables as independent. That step is wrong, and getting it wrong leads to badly mistaken conclusions.
The reason is exactly what correlation measures. Covariance, and so correlation, captures only the LINEAR association between variables — the tendency to lie along a straight line. Two variables can be uncorrelated (rho = 0, no linear trend) while being completely, even deterministically, dependent through some curved relationship. Independence is a far stronger demand: it requires the joint to factor for EVERY event, ruling out all forms of dependence, linear and nonlinear alike. So independence implies zero correlation, but not the reverse.
There is one famous exception that fuels the confusion: if (X, Y) is jointly (bivariate) normal, then uncorrelated does imply independent. People over-generalise from that special case. In general it is false — the honest rule is: independence gives you zero correlation for free, but zero correlation buys you only the absence of a straight-line trend, nothing more.
Let X be uniform on (-1, 1) and Y = X^2. Then E[X] = 0 and E[XY] = E[X^3] = 0, so Cov(X, Y) = E[XY] - E[X]E[Y] = 0 and rho = 0. Yet Y is a deterministic function of X — knowing X fixes Y exactly. Uncorrelated, but as dependent as it gets.
A clean counterexample: a parabola gives zero correlation with total dependence.
The one safe converse: if (X, Y) is jointly normal, uncorrelated does imply independent. Outside that special case, never infer independence from rho = 0.