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arXiv · 2609.23820

Using the Blackwell-Cover Guessing Strategy to Separate Unobserved Bernoulli Trials

James Stein
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Abstract

Blackwell and Cover both proposed the same method for guessing the larger of a pair of two numbers when one can only observe one and the other of the two numbers remains unobserved. As originally stated, the procedure ended with a computation showing that the probability of correctly guessing the larger number was greater than one-half. In this paper, we investigate what we can say when about the number in the pair that was not observed we employ the Blackwell/Cover procedure. Let α1 and α2 be two different numbers in (0,1]. Let G be the uniform distribution on [0, α1] and H be the uniform distribution on [0, α2]. We think of G and H as being represented by the heads probabilities of coins. We use a variant of the Blackwell-Cover strategy ro take pairs of coins chosen from distributions created from G and H such that one of the coins is flipped, and as a result of that flip the unobserved (unflipped) coin is placed in one of two sets A and B. We show that there are situations in which this can be done so that the mean of the heads probabilities of the unobserved coins in A is less than the mean of the heads probabilities of the unobserved coins in B. We show that there are situations in which recursive application of this process results in a set consisting almost completely of coins from one of the two distributions G and H, and another set consisting almost completely of coins from the other distribution.

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James Stein. 2026-09-20. Using the Blackwell-Cover Guessing Strategy to Separate Unobserved Bernoulli Trials. https://arxiv.org/abs/2609.23820

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