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Consider a gambler who on each bet either wins 1 with probability p or loses 1 with probability q=1-p, with the results of successive bets being independent. The gambler will stop betting when they are either up k or down k. Letting N be the number of bets made, we show that N is a new better than used random variable. Moreover, we show that if k is even then N/2 has an increasing failure rate, and if k is odd then (N+1)/2 has an increasing failure rate. 

Consider a set of n players. We suppose that each game involves two players, that there is some unknown player who wins each game it plays with a probability greater than 1/2, and that our objective is to determine this best player. Under the requirement that the policy employed guarantees a correct choice with a probability of at least some specified value, we look for a policy that has a relatively small expected number of games played before decision. We consider this problem both under the assumption that the best player wins each game with a probability of at least some specified value >1/2, and under a Bayesian assumption that the probability that player i wins a game against player j is its value divided by the sum of the values, where the values are the unknown values of n independent and identically distributed exponential random variables. In the former case, we propose a policy where chosen pairs play a match that ends when one of them has had a specified number of wins more than the other; in the latter case, we propose a Thompson sampling type rule. 

We develop recursions for computing the mean and variance of the number of pooled tests needed by the halving scheme to determine the disease positive members of a population. It is assumed that each population member is independently positive with probability p, that the individual blood samples can be pooled to test whether or not at least one member of that pooled group is positive, and that a positive tested group is then split in half and the process continued. 

Abstract We consider the Bernoulli bandit problem where one of the arms has win probability α and the others β, with the identity of the α arm specified by initial probabilities. With u = max(α, β), v = min(α, β), call an arm with win probability u a good arm. Whereas it is known that the strategy of always playing the arm with the largest probability of being a good arm maximizes the expected number of wins in the first n games for all n , we conjecture that it also stochastically maximizes the number of wins. That is, we conjecture that this strategy maximizes the probability of at least k wins in the first n games for all k , n . The conjecture is proven when k = 1, and k = n , and when there are only two arms and k = n - 1.