Let's deal with spin $1/2$ systems and fix a value (e.g. $+1/2$) in a given direction (e.g. $z$). Following a measurement along an orthogonal direction (e.g. $x$), we obtain 50% probability for $+1/2$ and 50% for $-1/2$. Fine.
Let's do the same with spin-one systems and fix the value $+1$ in the $z$ direction. When measuring the $x$ component, the probabilities turn out to be $1/4,1/2,1/4$ and not $1/3,1/3,1/3$ as one would naively extrapolated from spin one-half case.
Besides doing the correct calculation, is there any heuristic/intuitive argument to explain the difference?