Past exam of the options I for mathematics and philosophy course of the University of Oxford / 2013 / N / 1 / c

The five specified masses sum to one, so there are no other possible outcomes. Symmetry gives $\mathbb{E}X=0$, while $\mathbb{E}Y=1/4$. The two terms with $Y=1$ have opposite $X$ values and equal masses, soNevertheless $\mathbb{P}(X=0,Y=1)=0$, whereas $\mathbb{P}(X=0)\mathbb{P}(Y=1)=(1/2)(1/4)=1/8$. Therefore $X$ and $Y$ are not independent. They are uncorrelated random variables, illustrating that zero covariance alone does not imply independence.