Play-the-Winner sampling for a fixed sample size binomial selection problem

In a sampling problem for selecting the better of two binomial populations with a fixed total sample size, two sampling rules are shown to have equal probabilities of correct selection. One of them is the so-called Play-the-Winner sampling rule and the other is the usual Vector-at-a-Time sampling rule. As a corollary it is also shown that for the Play-the-Winner rule the expected number of observations on the poorer treatment, i.e. on the population with smaller p, is never greater than for the Vector-at-a-Time rule.