  Given an $r$-uniform hypergraph $H=(V,E)$ and a weight function
$\omega:E\to\{1,\dots,w\}$, a coloring of vertices of $H$, induced by $\omega$,
is defined by $c(v) = \sum_{e\ni v} w(e)$ for all $v\in V$. If there exists
such a coloring that is strong (that means in each edge no color appears more
than once), then we say that $H$ is strongly $w$-weighted. Similarly, if the
coloring is weak (that means there is no monochromatic edge), then we say that
$H$ is weakly $w$-weighted. In this paper, we show that almost all 3 or
4-uniform hypergraphs are strongly 2-weighted (but not 1-weighted) and almost
all $5$-uniform hypergraphs are either 1 or 2 strongly weighted (with a
nontrivial distribution). Furthermore, for $r\ge 6$ we show that almost all
$r$-uniform hypergraphs are strongly 1-weighted. We complement these results by
showing that almost all 3-uniform hypergraphs are weakly 2-weighted but not
1-weighted and for $r\ge 4$ almost all $r$-uniform hypergraphs are weakly
1-weighted. These results extend a previous work of Addario-Berry, Dalal and
Reed for graphs. We also prove general lower bounds and show that there are
$r$-uniform hypergraphs which are not strongly $(r^2-r)$-weighted and not
weakly 2-weighted. Finally, we show that determining whether a particular
uniform hypergraph is strongly 2-weighted is NP-complete.
