An Instance-Based Algorithm for Deciding the Bias of a Coin

TL;DR

This paper introduces an instance-based algorithm that efficiently determines whether an unknown biased coin's probability of heads is greater or less than a specified value, with high confidence and minimal flips.

Contribution

The paper presents a novel algorithm that decides the bias of a coin relative to a threshold with optimal expected flips, improving efficiency over previous methods.

Findings

Expected flips are logarithmic in inverse error and confidence levels.

Algorithm guarantees decision correctness with probability at least 1 - δ.

Efficiency scales with the inverse square of the bias difference.

Abstract

Let $q \in (0,1)$ and $\delta \in (0,1)$ be real numbers, and let $C$ be a coin that comes up heads with an unknown probability $p$, such that $p \neq q$. We present an algorithm that, on input $C$, $q$, and $\delta$, decides, with probability at least $1-\delta$, whether $p<q$ or $p>q$. The expected number of coin flips made by this algorithm is $O \left( \frac{\log\log(1/\varepsilon) + \log(1/\delta)}{\varepsilon^2} \right)$, where $\varepsilon = |p-q|$.