For several pairs (P, Q) of classical distributions on ℕ0, we show that their stochastic ordering P ≤stQ can be characterized by their extreme tail ordering equivalent to P({k*})/Q({k*}) ≥ 1 ≥ limk→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 (bn1,p1 ≤stbn2,p2 if and only if n1 ≤ n2 and (1 - p1)n1 ≥ (1 - p2)n2, or p1 = 0), both negative binomial (b−r1,p1 ≤stb−r2,p2 if and only if p1 ≥ p2 and p1r1 ≥ p2r2), 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).

The distribution of a fixed sum of independent and identically distributed random variables with the modified geometric distribution is the same as the distribution obtained by the compounding by a binomial distribution of either a negative binomial distribution or a Pascal distribution. This result can be used to obtain three summations for the game score probabilities of a two-person game, and leads to the consideration of various ways of dividing up the trials of the game. The game score probabilities are then used to consider the ‘fairness’ of four games and to analyse various methods of ‘setting’ (or ‘tie-breaking’) the games.