For several pairs (P, Q) of classical distributions on ℕ0, we show that their stochastic ordering P ≤st
Q can be characterized by their extreme tail ordering equivalent to P({k
*})/Q({k
*}) ≥ 1 ≥ lim
k→k
*
P({k})/Q({k}), with k
* and k
* denoting the minimum and the supremum of the support of P + Q, and with the limit to be read as P({k
*})/Q({k
*}) for finite k
*. This includes in particular all pairs where P and Q are both binomial (b
n
1,p
1
≤st
b
n
2,p
2
if and only if n
1 ≤ n
2 and (1 - p
1)
n
1
≥ (1 - p
2)
n
2
, or p
1 = 0), both negative binomial (b
−
r
1,p
1
≤st
b
−
r
2,p
2
if and only if p
1 ≥ p
2 and p
1
r
1
≥ 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 Lévy 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).