Bernoulli factory: The 2𝚙-coin problem

Abstract This paper aims to address the Bernoulli factory problem of <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mn>2</m:mn> <m:mo>⁢</m:mo> <m:mi mathvariant="monospace">p</m:mi> </m:mrow> </m:math> 2\mathtt{p} coins by analysing the relationship between the negative binomial distributions and binomial distributions generated on the same chain of coin flips. The proposed algorithm requires fewer conditions on the constructed sequences compared with the existing algorithms. The feasibility of obtaining such <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mn>2</m:mn> <m:mo>⁢</m:mo> <m:mi mathvariant="monospace">p</m:mi> </m:mrow> </m:math> 2\mathtt{p} -coin based on 𝚙-coins will be considered as well.