  For several pairs $(P,Q)$ of classical distributions on $\N_0$, we show that
their stochastic ordering $P\leq_{st} Q$ can be characterized by their extreme
tail ordering equivalent to $ P(\{k_\ast \})/Q(\{k_\ast\}) \le 1 \le \lim_{k\to
k^\ast} P(\{k\})/Q(\{k\})$, with $k_\ast$ and $k^\ast$ denoting the minimum and
the supremum of the support of $P+Q$, and with the limit to be read as
$P(\{k^\ast\})/Q(\{k^\ast\})$ for $k^\ast$ finite. This includes in particular
all pairs where $P$ and $Q$ are both binomial ($b_{n_1,p_1} \leq_{st}
b_{n_2,p_2}$ if and only if $n_1\le n_2$ and $(1-p_1)^{n_1}\ge(1-p_2)^{n_2}$,
or $p_1=0$), both negative binomial ($b^-_{r_1,p_1}\leq_{st} b^-_{r_2,p_2}$ if
and only if $p_1\geq p_2$ and $p_1^{r_1}\geq p_2^{r_2}$), or both
hypergeometric with the same sample size parameter. The binomial case is
contained in a known result about Bernoulli convolutions, the other two cases
appear to be new.
  The emphasis of this paper is on providing a variety of different methods of
proofs: (i) half monotone likelihood ratios, (ii) explicit coupling, (iii)
Markov chain comparison, (iv) analytic calculation, and (v) comparison of Levy
measures. We give four proofs in the binomial case (methods (i)-(iv)) and three
in the negative binomial case (methods (i), (iv) and (v)). The statement for
hypergeometric distributions is proved via method (i).
