This calculator uses the following formulas to compute sample size and power, respectively:
$$
n_A=\kappa n_B \;\text{ and }\;
n_B=\left(\frac{p_A(1-p_A)}{\kappa}+p_B(1-p_B)\right)
\left(\frac{z_{1-\alpha/2}+z_{1-\beta}}{p_A-p_B}\right)^2$$

$$1-\beta=
\Phi\left(z-z_{1-\alpha/2}\right)+\Phi\left(-z-z_{1-\alpha/2}\right)
\quad ,\quad z=\frac{p_A-p_B}{\sqrt{\frac{p_A(1-p_A)}{n_A}+\frac{p_B(1-p_B)}{n_B}}}$$
where

- $\kappa=n_A/n_B$ is the matching ratio
- $\Phi$ is the standard Normal distribution function
- $\Phi^{-1}$ is the standard Normal quantile function
- $\alpha$ is Type I error
- $\beta$ is Type II error, meaning $1-\beta$ is power