The double biased coin adaptively allocates new subjects to one of two study arms, usually denoted treatment and control. This method works by adapting the probability a new subject is allocated to either arm based on the success of previous subjects in that arm.

There are two methods that this calculator supports for adapting the probability a subject is assigned to either arm: 'minimization' and 'urn'. Let \(n_{t}\) be the number of treatments, of which \(s_{t}\) were successful, and \(n_{c}\) be the number of controls, of which \(s_{c}\) were successful. We can define \(p_{t} = \frac{S_{t}}{N_{t}}\) and \(p_{c} = \frac{S_{c}}{N_{c}}\) as the probability a subject is successful in the treatment and control arms respectively. Using the minimization method, the probability a new subject is allocated to the control arm is given by:

\[ \frac{\sqrt{p_{c}}}{\sqrt{p_{c}} + \sqrt{p_{t}}}. \]Using the urn method, the probability a new subject is allocated to the control arm is given by:

\[ \frac{1 - p_{t}}{(1 - p_{c}) + (1 - p_{t})}. \]- Wei, L.J. & Durham, S. (2015), "The Randomized Play-the-Winner Rule in Medical Trials,"
*Journal of the American Statistical Association,*73:364, 840-843.